I came across the phrase in this article:
And "in this case, the law's terms ensure that, when the federal government seeks a procedural advantage against an individual, it will at least supply him with a single and reasonably comprehensive statement of the nature of the proceedings against him. If men must turn square corners when they deal with the government," Gorsuch declared, "it cannot be too much to expect the government to turn square corners when it deals with them."
It seems to deal with matters of compunction in public service, e.g. following rules and procedures to the best of one's ability. Another source seems to confirm this, citing a little-known "square corners doctrine." But what does it actually mean, and where does it come from?
My hypothesis is that it's a cousin to "make sure your 't's are crossed and your 'i's are dotted," but catchier and more efficient.
EDIT It seems it could be the opposite of "cutting corners," though closer study of THAT phrase's origin (whether the cater-corner or the hunting explanation) seems to indicate that this is an incomplete answer.