This is slang for "a rigidly conventional person": "Get with it; don't be a square." Trite phrases and words to be wary of: "square peg in a round hole," "on the square," "square shooter," "square off' ("assume a fighting position"),"square up" ("to settle," "to pay"),"square the circle" ("to attempt the impossible"),"square with one's conscience," "square meal," and "square deal."
Dictionary of problem words and expressions. Harry Shaw. 1975.