Green’s Dictionary of Slang

1. to placate or conciliate someone, to apologize.

1926		J. Doone Timely Tips For New Australians 23: TO ‘SQUARE OFF.’—To smooth over the resentment of another.
1941		Baker Popular Dict. Aus. Sl.
1955		D. Niland Shiralee 129: Just so you can square off. Just so you can be a good feller with me and a good feller with yourself.
1963		‘Nino Culotta’ Gone Fishin’ 158: He thinks you put him in to the Inspector. You’d better go and square off.
1982		G. Newbold Big Huey 157: When he tried to square off and asked us if we’d like to knock off early one day, we told him we’d rather work.
2001		D. Looser Boobslang [U. Canterbury D.Phil. thesis] 175/2: square off v. 1 to apologise, to make excuses [...] 2 to justify oneself in an argument, to give one’s side of the story.
2011		J.J. Connolly Viva La Madness 282: Mort’ll square it off — a few quid compo, a few rounds of drinks.

2. to pay; can be used quite legitimately, but often carries a sense of corruption, bribery etc.

1924		‘Digit’ Confessions of a Twentieth Century Hobo 92: I squared off the maitre d’hotel.
1963–74		B. Humphries Barry McKenzie [comic strip] in Complete Barry McKenzie (1988) 105: I’ll square off this Arab.
1985		B. Humphries Traveller’s Tool 14: When she’s squaring off the milkman in the stained old pink brunch coat.

3. to pay off, e.g. a debt.

1930

‘Leslie Charteris’ Enter the Saint 26: I’ll get you enough out of it to square off your debts.

4. (N.Z. prison) to get revenge.

2001

D. Looser Boobslang [U. Canterbury D.Phil. thesis] 175/2: square off v. 4 to exact revenge.