Mr. Square is the second tutor for Tom and Mr. Blifil when they are children together. Squire Allworthy hires Mr. Square because he believes that Mr. Square's philosophy will balance out Mr. Thwackum's puritanical religious faith. Mr. Square studies Greek thinkers like Plato and Aristotle, who give him a lot of abstract ideas about virtue and living a decent life. (For more on Mr. Square's philosophy, check out our list of "Allusions," under "Plato.")
But in fact, it turns out that Mr. Square and Mr. Thwackum spend all of their time squabbling about religion vs. philosophy. They are so interested in abstract theoretical ideas about good and evil that they totally fail to teach Tom and Mr. Blifil anything about practical morality or ethics. It's lucky that Tom seems to have been born with good instincts, or who knows what he would have learned from these two.
While Mr. Square joins Mr. Thwackum in hating on Tom and bad-mouthing him to Squire Allworthy, he is a better guy overall than Mr. Thwackum. He's not violent. And he also has a deathbed change of heart in Book 18, when he writes to Squire Allworthy to confess that he saw Mr. Blifil turning Squire Allworthy against Tom unfairly. Mr. Square testifies that, in fact, Tom has always loved and respected Squire Allworthy dearly, while Mr. Blifil is a lying little worm. (He doesn't use exactly those words, but that's what he means.)
At first, Mr. Square and Mr. Thwackum appear to be at opposite sides of the same problem, since they are both interested in Goodness as an abstract thing. But as the book continues on, we come to like Mr. Square a little more. He's still kind of lame, since he only decides to seek forgiveness when he thinks that he's about to die and starts wondering about Heaven, Hell, and the afterlife. Yet, even if Mr. Square regrets his bad treatment of Tom for selfish reasons (since he wants to go to Heaven), he still eventually tells Squire Allworthy the truth.
The character difference between the two men really comes out once Squire Allworthy receives letters from both of them at the same time, in Book 18, Chapter 7. Mr. Square writes to ask Squire Allworthy's forgiveness for his poor treatment of Tom. And Mr. Thwackum writes scolding Squire Allworthy for liking Tom and demanding a second job. Even though both Mr. Square and Mr. Thwackum make Tom's life a misery when he's younger, Mr. Square redeems himself to some extent.
We get a sense that Mr. Square is going to turn out to be superior to Mr. Thwackum in the end because he shares Tom's particular vices. When Tom decides that he has to give up his relationship with Molly Seagrim in order to pursue Sophia Western, he surprises Mr. Square in Molly's bedroom. While it's bad that Mr. Square preaches about abstinence while still jumping into bed with Molly, the fact that he has human failings and weaknesses makes him more sympathetic than Mr. Thwackum. By contrast, Mr. Thwackum really is as austere and brutal as he seems—which makes him a bully right to the end.