Green’s Dictionary of Slang

squarehead n.1

[square adj. (2) + -head sfx (1)]

1. an honest person, a respectable person.

[Aus]J. Armour Diggings, the Bush, and Melbourne 16: I learnt that nearly all the company had been ‘Government men,’ as convicts style themselves. [...] [and] being a ‘square head,’ that is one outside of their community, I would readily be suspected if were tales told.
[Aus]Truth (Sydney) 12 May 5/7: I shouldn’t have anticipated much less luck for a ‘Square-head’ unfortunate to get into the clutches of such a cute and foxy trickster.
[Aus]Sydney Sportsman (Surry Hills, NSW) 10 Oct. 1/1: Square heads won’t squeal at my notes, and as for the cronk crew, I like to hear them squeak.
[Aus]K. Tennant Foveaux 312: ‘What’s y’r old man do for a crust?’ Curly asked. ‘Works,’ Neicie answered shortly. ‘Drives a taxi.’ Curly raised his eyebrows. ‘A squarehead.’.
[Aus]Sun. Mail (Adelaide) 25 Sept. 45/7: I always sought rhe company of ‘square heads’, men who have had no trouble with the law.
[Aus] ‘Whisper All Aussie Dict.’ in Kings Cross Whisper (Sydney) xxxix 4/5: square head: A conformist.
[Aus]R. Aven-Bray Ridgey-Didge Oz Jack Lang 45: Squareheads Non-criminals.
[Aus]R.G. Barrett Real Thing 99: Norton, being a bit of a square-head, Queensland country boy [etc.].
see sense 2.
[Aus]M.B. ‘Chopper’ Read Chopper 4 15: I’ve never really mixed with squareheads and normal people.

2. a stupid person.

[US]Van Loan ‘No Business’ in Taking the Count 149: He handed that big squarehead from the revenue cutter an awful trimming.
[US]P. Kyne Cappy Ricks 20: I’ll dig up a bosun among the squareheads.
[US]P. Ernst ‘The Chair Cheater’ in Phantom Detective Jan. [Internet] You squarehead—we’ve only got one coffin aboard, ain’t we?
[UK]B. Hill Boss of Britain’s Underworld 89: I was not like that bunch of swedes and squareheads they had in there from provincial towns.
[Aus]F.J. Hardy Four-Legged Lottery 86: Look who’s here, the square head from the library.
[US]H. Rawson Dict. of Invective (1991) 372: The older squarehead, a stupid person.
[Aus]Tupper & Wortley Aus. Prison Sl. Gloss. [Internet] Squarehead. A naive or uncorrupted person. Thus someone doing prison for the first time who doesn’t understand the prison values and lives according to the values of society outside: or a straight (non-gaol) person.

3. (Aus.) a timid or conscience-ridden thief.

[Aus]Baker Popular Dict. Aus. Sl.

4. (Aus.) one who has no previous criminal convictions.

[Aus]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.
[Aus] ‘Whisper All Aussie Dict.’ in Kings Cross Whisper (Sydney) xxxix 4/5: square head: [...] An unconvicted person.