Green’s Dictionary of Slang

1. (UK Und.) the laws and customs of the underworld, by which all criminals should abide.

1812		Vaux Vocab. of the Flash Lang. in McLachlan (1964) 268: square: all fair, upright, and honest practices, are called the square, in opposition to the cross.
1846		Swell’s Night Guide 60: The Nummer spouts the tick and linker for quarteroo cooter – did the square with the kids.

2. a respectable way of life.

1848		Bell’s Life in Sydney 26 Feb. 1/4: Then you mean to cut the cross, and work the square.
1865		Dickens Our Mutual Friend (1994) 579: The best amends for having got out of the square is to get back into the square.
1965		B. Jackson Get Your Ass in the Water (1974) 86: I tried to put the square on them but they looked at me and laughed.

3. (US black) a trustworthy individual.

1939

Louis Jordan ‘’Fore Day Blues’ 🎵 Hmmm, I can’t get started / Since you threw me down, / I thought you were the square, babe, / And you played me for a clown.