I think this expression is perhaps one that Harper Lee may have heard in Alabama, where she grew up. In 1985, twenty-five years after the publication of To Kill A Mockingbird, Ronald Reagan made a joke about a three-legged chicken during a speech to the National Association of Realtors. Reagan was born and raised in Illinois, not too far from Alabama, and so perhaps this expression originates from somewhere around that region of the country.
In the novel, Atticus says that Judge John Taylor looked at Bob Ewell (during the trial of Tom Robinson) as if he were a "three-legged chicken or a square egg." The implication is that Bob Ewell's testimony was nonsensical. In the story, it is implied that Ewell was the one who raped his own daughter, Mayella; and yet in the trial, he testifies that it was Tom Robinson.
When recounting Judge Taylor's confused, incredulous expression, Atticus also says, "Don't tell me judges don't try to prejudice juries." The implication here is that Judge Taylor was looking at Bob Ewell as if he were "a three-legged chicken or a square egg" to communicate to the jury that Ewell's testimony was nonsensical.