Green’s Dictionary of Slang

1. honest, respectable.

1857		T. Hughes Tom Brown’s School-Days (1896) 72: We was six miles from the town, when we meets an old square-headed gray-haired yeoman chap, a jogging along quite quiet.
1922		Joyce Ulysses 80: Squareheaded chaps those must be in Rome: they work the whole show.

2. stupid; naive.

1914		Sport (Adelaide) 13 Nov. 5/6: Joe S., the square-headed Turk, ought to wash his socks.
1918		‘Max Brand’ ‘Above the Law’ in Coll. Stories (1994) 28: You big, clumsy, square-headed, bat-eyed fool!
1922		Joyce Ulysses 80: Squareheaded chaps those must be in Rome: they work the whole show.
1925		G.H. Mullin Adventures of a Scholar Tramp 39: The barkeeper was a large, square-headed Pole.
1959		Belfast Teleg. 22 Aug. 3/8: The West German newspaper [...] criticised President de Gaulle — ‘the squareheaded man in Paris’.
1992		R.G. Barrett Davo’s Little Something 47: He was still a bit of a square headed country boy at heart.