What is the origin of the phrase "zero, zip, zilch, nada"?

These are all called Squatitives, believe it or not.
In particular, zero, zip, zilch (and perhaps nada) are Bahnhofers.

These words participate in a lot of odd syntax involving emphatic negation. For instance, both kinds of sentence below occur, and are intended to mean the same:

• He doesn't know Zero/Zilch/Zip/Nada about it.
• He knows Zero/Zilch/Zip/Nada about it.

There are many more, in many languages, and many more constructions involving them. A short list of Squatitives currently being researched is available here.

As a translator I now have reason to believe that the term "zip", at least, was only just coming into wide use around the early or mid-1970s. I have a Slovenian translation of Bob Woodward and Carl Bernstein's sequel to "All the president's men", called "The final days". At one point in 1974 the Supreme Court handed down a unanimous (8-0) ruling that Nixon must release all the tapes - a major blow to his hopes of staying in power. On hearing the news, one of his White House aides rushed past his secretaries saying - in the Slovenian translation - "Osem - čik!" The first word means "Eight", and the second is a vague sound that could be taken as meaning "clunk!", "bang!" or "thump!" (its only real meaning is "cigarette butt", but that would mean nothing here). One of the secretaries called after him "Kaj pomeni 'čik'?" - "What does 'čik' mean?" This got me wondering what the aide had said in English. I didn't have a copy of the original English version, but soon found the relevant passage on the Internet: "Eight - zip!" (clearly referring to the 8-0 ruling), to which the secretary replied "What does 'zip' mean?" I concluded that by 1974 some Americans were already using "zip" in the meaning "nothing", but that some other Americans didn't yet know that meaning - also that the Slovenian translator was not (yet) familiar enough with English to pick up the new meaning, and had mistaken "zip" for a vague sound, hence the translation "čik" rather than some word for "nothing". Working in the former Yugoslavia in the mid-1970s he also might not have had much contact with the latest American slang - and he also needed to account for why the secretary had failed to understand her boss, so a word with a clear meaning would not have made sense. All this would seem to pin down the rise and spread of "zip" = "nothing" to the early or mid-1970s.

On reflection, it's also possible that the translator did understand the new meaning of "zip" but was unable to find a contemporary Slovenian slang word for "nothing" that some people would understand and but others would not - so, to account for the secretary's failure to understand her boss, he turned "nothing" into a meaningless sound that would leave her puzzled, forcing her to ask what it meant. If so, this was a very skillful piece of translation! But he could simply have omitted her question - so I'm inclined to think that, like the secretary in Washington, he really didn't understand what "zip" meant.

In any case, it still looks as if the meaning of "zip" was very new at the time in English.

Related synonym strings from 1953–1974

Hugo's excellent answer covers a 1978 occurrence of the full-blown phrase "zero, zilch, zip, nada, nothing" as well as a couple of earlier and shorter variants. There are, however, a number of earlier instances—from the period 1953–1974—in which writers or speakers use a string of three or four synonyms for nothing to similar effect.

Here are some instances that searches of the Google Books, Hathi Trust, and Elephind databases turn up.

From Frederic Wakeman, Mandrake Root (1953) [combined snippets]:

It didn't seem fair, considering the nilness of the miller's love-offering to his adoring daughter, an offering which was nothing, zero, nada; and comparing that cipher to the magnitude of the husband's hunger for the wife who never had been able to flee her father's nest.

From Herbert Gold, "The Beat Mystique" (1958), reprinted in Cesar Grana, On Bohemia: The Code of the Self-exiled (1990):

Let us now move in closer to the hipster's harried heart. When the hipster makes it with a girl, he avoids admitting that he likes her. He keeps cool. He asks her to do the work, and his ambition is to think about nothing, zero, strictly from nadaville, while she plays bouncy-bouncy on him.

From George Garrett, "Love Is a Cold Kingdom," in Shenandoah, volume 12 (1960–1961) [combined snippets]:

And Angus, damn him. He was the boy with talent, with all the talent, my idea of a poet. Married and divorced. The Army. Sixty days bad time in the stockade for AWOL. Wounded in Korea. Back to the States. Work for a publisher. Fired. Work for a magazine. Fired. Teaching at a boy's prep school. Nervous breakdown. Back to graduate school. Next the State kooky hatch for a couple of years. Poems? Zero, zip, none. And now he was out and in the city and had called me from the station that he was coming to see me.

From an untitled item in Theatre Organ: Journal of the American Association of Theatre Organ Enthusiasts, volumes 1–5 (1959[?]):

Response to the request for information on home installations (see THEATRE ORGAN, Vol. 1, No. 3 & 4, the double issue, Page 4 ) has been exactly zero -- 'nothing -- blank!!! Surely there beats within the breast of at least ONE theatre organ owner enthusiast a heart full of ever-flowing with pride of ownership! We have had many excellent pictures sent in with letters of general context, but in not one case have we had enough information for a story.

From Harold Sullivan, Trial by Newspaper (1961) [combined snippets]:

What did the judiciary of Massachusetts do about the destruction of the constitutional guarantee of a fair and impartial trial in the case of Woodward, a fifteen-year-old child when arrested, and an acknowledged mental defective?

Zut, zero, nothing.

From an unidentified play in The Chicago Review, volume 16 (1963) [combined snippets]:

STELLA: (stopped by that question) I don't know. But I should've at least tried to do something.

VITO: I'll tell you what you coulda done. Nothing. Zero. Goose egg. You showed good sese. It ain't your bag of oats, so you kept your nose out.

From an advertisement for Blue Cross in the [East Orange, New Jersey] Catholic Advocate (December 8, 1966):

Blue Cross has 76 boards of directors, and their rosters read like the financial Who's Who of the Western World.

And we pay them nothing. Zero. Naught.

In fact, it probably costs them plenty of money to give us their time and attention.

From "Roger Simon, "O Union, My Union," in the [Illinois] Daily Illini (November 12, 1968):

In black and white the Union prints in yet another publicity booklet titled "Why We Are Here": "If the job of the Illinois Union can be summed up in any single sentence, it is, 'to be of service to the students, faculty, staff, guests and alumni of the University.' This service is our prime, overriding purpose."

Well, fine and good. But those cynics among toy ask, "How much then, my man, do the faculty and staff pay for the use of the Union. (Guests to remain guests should come in free.) The answer comes back: Nada. Nothing. Zero. Zip. Yes, it is spelled out in the Undergraduate Study manual: "the fee is waived for all staff members, academic, administrative, or permanent non-academic staff members or allied agencies on appointment for at least 25 per cent of full time."

From U.S. Congress, Hearings Before the Subcommittee on Economy in Government of the Joint Economic Committee, Congress of the United States (December 20, 1972):

[Chairman PROXMIRE] ... Only 5 out of 34 contractors disclosed anything about cost overruns, only one disclosed its delivery problems, only one disclosed its claims against the Government, and none—zero, zip—disclosed the problem of estimating costs in defense contracts. Does not this survey show rather conclusively that the major defense contractors are failing to disclose the facts about their military work?

Robert Chapman, New Dictionary of American Slang (1986) reports that zilch in the sense of "nothing" entered U.S. English in the late 1960s, although an antecedent may be found in the 1930s term "Joe Zilch":

zilch 1 n fr late 1960s Nothing; zero; =ZIP. ... in British use, but not US, zilch be reinforced by nil "zero"; all senses may derive fr the early-20th-century US college use Joe Zilch "any insignificant person," popularized during the 1930s by ubiquitous use in the humor magazine Ballyhoo with the spelling Zilch, an actual German surname of Slavic origin; ...

In any event, given the late arrival of zilch as a stand-alone synonym for zero, it is not surprising that it doesn't appear in any relevant synonym strings before the early 1970s.

From U.S. House of Representatives, The Credit Crunch and Reform of Financial Institutions: Hearings Before the Committee on Banking and Currency (September 10, 1973):

Mr. REUSS. Let me try to get you to see it. Your millionaire who is already getting all of these bonanzas under your bill is going to get another one; he deducts the $750, and since he is in the top bracket, that means he gets, cash in pocket, $525. Your little person, like the widow whose case I put, gets zero, zilch, nothing. I urge your excellent organization to sit down with counsel and rethink that one. I hope that your recommendations, when we next see them, will be purged of what was well meaning but really a disaster, because it would put in another big loophole. So give it some thought, as I know you will.

From Richard Biow, "Enjoy Your Bible," in the Pittsburgh [Pennsylvania] Catholic (July 19, 1974):

The Christian has absolutely no idea of "sin" that does not mean "disobedience of the covenant." In this sense we have no "ethics," no "nice," no "naughty." We either carry out our side of the covenant or we do not.

Please note — and make everybody in hearing-range note — that this means nothing — zero, zilch — outside of the community of the covenant. For example: God promises to make our community "numerous as the stars." Even God can't make an individual person "numerous"!

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Synonym strings signifying 'nothing' from 1852–1919

Although the examples I've cited thus far have come from the 1950s, 1960s, and 1970s, earlier instances of strings of three or four synonyms signifying "nothing" do occur. These tend to sound considerably different from the modern "zero, zilch, zip, nada, nothing" formulation—and that difference, together with their much greater age—may indicate that they are not connected by any direct lineage. Nevertheless, these very early instances represent a kind of prehistory of the current phrase, and they suggest that English speakers find something appealing and rhetorically effective in such assemblies of synonyms.

From an untitled news item in the [Brazoria] Texas Planter (July 28, 1852):

News.——Our files of exchanges come to us almost utterly devoid of news. Nothing but a continual jargon of politics sufficient to completely and effectually confuse stronger heads than ours. We overhauled and read with what of patience and perseverance we were possessed of for two hours, and for a result we had a grand total of nothing, 0, zero infinitessimy—and a most miserable headache. We advise our friends to abjure political newspapers till after November, if they have regard for their personal comfort and well being.

From "The Thermometer," in the Cambridge [Massachusetts] Chronicle (March 12, 1868):

Putting tho bulb into the snow and suit, he [Fahrenheit] marked the point to which the mercury, which fills the tube, fell, 0 (cipher), zero, nothing; because, as before said, he thought the amount of heat was nothing. But had he lived in our cold New England, instead of Germany, ho would have found, perhaps to his sorrow, that our wintry air is often colder than his ice and salt.

From "George and the Meter," in the Wheeling [West Virginia] Daily Intelligencer (March 8, 1898):

One day a plumber for the gas company brought around a meter and told George to connect it. George said he would, but other duties pressed him so hard that he did not give it any attention for several weeks, and it lay unconnected and undisturbed in a dark closet of the castle. When the meter came it registered nit, naught: zero, nothing. But he noticed that during the time it was in a dark closet It registered 4-11-44. Judge of his surprise!

From Philip Firmin, "Our Telephone System," in the Los Angeles [California] Herald (April 25, 1901):

Blockleaf—Now please don't get try to me excited. I am endeavoring to be calm. Won't you listen to this number: Main—not Green nor Blue, nor ashes of roses, nor mouse-color, but Main—same as Main street—sabe? Now for the numerals: Two, do you understand—yes, that's right (excitedly) she is getting it, Greenstone—two and then double one, eleven, one-one—yes, that's right; and then nought—yes, ought If you prefer, or zero, nothing, cipher; now, all together: Main, two, double-one, nought. Hooray! That's right.

And from "Fishermen Urged to Call Off Trips," in the Canon City [Colorado] Record (August 21, 1919):

Scores of vacationists returned during the last few days from the streams in this section. The rains caused the creeks and other waterways to rise rapidly and become muddy. Their total of catches amounted to naught, nothing, zero, non est at nil.

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Conclusions

The rhetorical strategy of invoking a thesaurus-like string of synonyms—rather than a single suitable word—to make a point seems peculiarly effective in the case of words that mean "nothing." At ant rate, writers and speakers have been using it for a long time. Although (as Hugo's answer notes) the exact wording "zero, zilch, zip, nada, nothing" may not go farther back than 1978, other synonym strings have been serving much the same purpose in connection with much the same underlying idea since at least 1852.

When a form of expression has that kind of staying power, I think it is reasonable to view it as wielding greater linguistic force than one might otherwise be inclined to grant it. Psychologically, a reader or listener pelted with multiple versions of the same idea is likely to feel the weight of the verbal overload—both as something faintly ridiculous in its excess and as something asserted with maximum vehemence.