Learn English – the meaning of “I got you”

A Google search for define:square gives me as one meaning

a formal and conservative person with old-fashioned views

which is what "square" means in your example.

The statement "your mom did a number on you" is a disparaging remark usually said by women who find some kind of flaw in the men with whom they have a relationship. The flaws may range from the man having an inability to commit to the relationship, a failure to have matured sufficiently (because of the overprotectiveness of the mother), or some other related issue.

It does not necessarily mean that the man had bad childhood memories. It's more of a commentary by a woman on how the training given by another woman (the man's mother) has unsuited the man for a proper relationship with an adult woman.