If there is only one square, it is quite natural to say "the square". If there are a number of ordered squares, it is quite natural to say "square one", "square two", "square three" and so on. It would also be natural to say "the first square", "the second square", "the third square", and so forth.
The ordinal numbers (first, second, third) are ordinary adjectives. They don't count as determiners. When the series of squares is definite, "the first square" makes sense. When the series of squares is indefinite, "a first square" makes sense.
The cardinal numbers (one, two, three) behave a bit differently. When they appear after a singular noun that they modify, they do count as determiners. This usage only makes sense when the series is definite. Phrases like "square one", "chapter one" or "job one" don't need the definite article. The trailing number is definite enough.
Cardinal numbers can also appear as weak determiners before a noun. In this case (and with a single obvious exception) they modify plural nouns: "one square", "two squares", "three squares", etc. I call these determiners weak because they don't prevent the use of the definite article. "The one square" can still sound natural. "A one square", however, sounds completely wrong.
The structure of "the first one" is much like the structure of "the first square". Here, the "one" is a pronoun modified by the adjective "first". The series continues "the second one", "the third one" and so on.
The same structure exists when "square" is an adjective. In that case, "the square one", "the round one", "the thin one", "the tall one" and the like all sound perfectly natural. Those are simply ordinary adjectives modifying an ordinary pronoun.
In the phrase "square one", the "one" is a modifier, specifically a determiner. That's the same role that the definite or indefinite article can fill, so no article is required. In the phrase "the square one", "one" is a pronoun. "Square" is a modifier but it's not a determiner. The determiner role is filled by the definite article.
None of this has anything to do with the fact that "square one" is an idiom for "the beginning", or that it is almost always paired with the word "back".