Green’s Dictionary of Slang

1. a timid or conscience-ridden thief.

1950

‘No. 35’ Argot in G. Simes DAUS (1993) 193: Squarie [...] a respectable person; also, a prisoner who has moral scruples (possibly corruption of old-fashioned ‘squaretoes’, a puntilious person).

2. one who has no previous criminal convictions.

1950		S.J. Baker in Sun. Herald (Sydney) 8 June 9/2: There’s [...] the ‘cleanskin,’ ‘squarehead’ or ‘squarey,’ the new crook who has had no previous convictions.
1953		Baker Aus. Speaks 124: squarey a crook who has not yet been convicted.