Of boughten the OED writes:

boughten, ppl. a. [irreg. f. bought ppl. a. by assimilation to foughten] = bought ppl. a. used poet. for the sake of metre, otherwise only dial. and in U.S. in application to purchased as opposed to home-made articles.

Under the OED’s entry for buy, in the section on that verb’s inflections and their historical spellings, at no point does anything even vaguely like boughten ever enter the picture for that verb’s past participle.

In Bryan Garner’s Oxford Dictionary of American Usage and Style (2000), he writes on page 51:

The form boughten (= store-bought as opposed to home-made) is an archaic past-participial adjective formed on the analogy of words such as broken, driven, and frozen. It still occurs in dialectal speech in the North and Northeast (in the sense “store-bought”), but not elsewhere. E.g.: “In those days any boughten [read store-bought] cookie we would see in Maine was made by the Loose-Wiles Biscuit Company, which had a huge brick factory near the railroad tracks in Boston’s North End” (Christian Science Monitor).

In Standard English boughten is never a past participle the way you have used it here. It is at most only ever a participial adjective, either in dialect speech or else in the United States for things you have purchased rather than made yourself.

In those dialects that still use it, boughten is usually just an adjective, not a past participle. It’s one of those oddities of a verb that presents a four-fold paradigm: the base verb, past tense, past participle, and then a distinct participial adjective ending in -en not seen in the past participle.

In this way it works somewhat like the verb drink works:

• drink, drank, drunk, drunken
• buy, bought, bought, boughten

So as adjectives we can say a drunken sailor or a store-boughten item, but those are still just have drunk and have bought when past participles.

Another example of this model can be found in U.K. speakers who wouldn’t be caught dead using gotten as a past participle even in the situations where North American speakers would use it, and yet those same U.K. speakers find nothing amiss with using it as a participial adjective in only-begotten son.

As the OED notes, this old participial adjective form of boughten is now reduced to dialects only or in the U.S. for a particular application. For example, it can still be heard in Cornwall and various places around the American Midwest.

But since Standard English isn’t so fond of it, if it is not already in your dialect, you should not use it, especially in formal writing. If you’re writing dialogue or just chewing the fat with friends, don’t worry about it.

My own rural dialect from the southeastern corner of Wisconsin retains boughten items, and I do use it in that fashion, but I don’t enjoy constantly having to justify this to people whose dialect lacks it and who are unaware of the word’s existence let alone its history.

On Correctness

In comments, the asker repeats the question about whether have boughten is “correct”, and whether there exists some definitive grammatical “rule” providing a simple yes or no answer respectively supporting or disputing the “correctness” of the have boughten construction.

If you’re a kindergarten teacher guiding your young charges into a normalized form of Standard English, then you will want to suggest to them that they not say that. Prescriptive normalization can have its place, and some would argue that this is one of them.

However, from a linguist’s perspective, the question doesn’t really make much sense. That’s because it presupposes tacit acceptance of the proposition that “correct” is even a thing, and moreover that it is a thing which is either black or white. That’s not how linguists describe the language they observe. It’s something of a category error that is apt to puzzle them; it’s the wrong question to ask them because it doesn’t make sense given how they look at language.

So-called grammatical “rules” are just codifications of collected observations about how real people actually speak. So grammatical just means “sounds ok to a native speaker”, while ungrammatical just means “doesn’t sound ok to a native speaker” — nothing more and nothing less. There can be infinitely many gradations of grey acceptability stretching between those two black-and-white fenceposts of yes and no.

The bottom line is that Standard English does not gladly admit have boughten, but various dialects do. If that’s your dialect, then so be it, but be warned that you may suffer the social opprobrium and stigmata commonly associated with dialect speech; check your wrists to make sure you’re still ok.