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.
1999		D. Looser ‘Boob Jargon’ in NZEJ 13 35: square off v. To apologise.
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.