The idiomatic expression appears to date back to the sixties:

square one (n.) the image of children’s board-games, e.g. snakes and ladders

(orig. US) the starting point, the beginning.

• 1960 [UK] Times 21 May 9/2: As far as building up a basis for profitable negotiations is concerned the two sides are back in square one.

• 1967 [US] N.Y. Times 18 June 4:1: The dilemma for the Kremlin is [...] thrown back to square one.

• 1971 [UK] Aberdeen Press 2 Apr. 22/9: The letter ‘put the situation right back to square one’.

• 1981 [UK] S. Berkoff Decadence in Decadence and Other Plays (1985) 37: You may lose him and start at square one again.

(GDoS)

The earliest print evidence of the use of 'square one' in the figurative sense, "the beginning, the starting point" (OED), has previously been thought to be in a 1952 book review published by The Economic Review. My research, however, uncovered a 1946 use in F. van Wyck Mason's Saigon Singer:

 "And now, mon Commandant," she said brightly, "I'm over my crise de nerfs. You may consider me back in Square One."
 "'Square One' in French sounds silly. Let's talk English." Though speaking lightly, he had definite reasons for making the suggestion.

Nothing of the intrinsic context offers a clue to the origin of the phrase, although the character's opinion that it "sounds silly" in French suggests it perhaps did not come from French.

The book reviewer's 1952 use of the phrase embeds a reference to an ancient board game called snakes and ladders:

Withal he has the problem of maintaining the interest of a reader who is being always sent back to square one in a sort of intellectual game of snakes and ladders.

The reviewer's analogy is sloppy, at best, and wildly inaccurate in fact, as shown by a 1904 illustration of the game board, and accompanying description. There, the game is called Moksha Patamu or "A Picture of Final Deliverance":

…You commence to play on square 36 — i.e., "Manhood's Estate."

So square 36, not square 1 is the starting point. Further, it becomes evident after study of the board and instructions that the only way to go 'back to square one' is to land on square 73, "Murder", where the snake causes a figurative death and rebirth as a tiger on square one.

Thinking the 1904 illustration might represent an unusual version of the game, I checked more than 100 other incarnations of the board, from 19th century Victorian England versions through the ridiculously toned-down 1943 US version (now called chutes and ladders, because, you know, we can't have those scary snakes causing death and rebirth in a children's game!), and found that although other versions of the game might start on square 1, no version incorporated multiple ways to return to square 1. At most, landing on only one of many possible squares returned the player to square 1.

These facts disqualify snakes and ladders as the origin of the figurative use of 'square one', in much the same way that "square 1" in the 1927 Radio Times visual key to broadcaster football commentary, not being the starting point of the game, disqualifies that origin.

Were it not for a niggling sense of unease caused by the semantic fluidity of the 'square one' metaphor, the facts would lead me to endorse hopscotch (versions) as a potential origin; however, I suspect that "square 1" is a natural metaphor for "the beginning" among English speakers conversant with basic arithmetic concepts. As such, no origin other than what is part and parcel of speaking the language, and understanding the significance of "1", was needed for the metaphor 'square 1' to emerge.

This is not to diminish the possible influence of hopscotch and simple board games, or other more esoteric sources such as the closed knight's tour, on the emergence and establishment of the now-common metaphor. Rather, it is to say that the 'square 1' metaphor stands, and is readily understood, without reference to any source or origin other than the meaning of the words and the number: number 1 is, after all the first number (when 0 is nothing), whether in a square or out of one.