Absolutely nothing beyond the definition. The complex plane is just R x R with some additional operations on it, which are irrelevant by your definition. The function doesn't have to be differentiable anywhere, continuous anywhere, or integrate to zero anywhere other than on the edges or on the main diagonals. Sorry. If you limited the question to say, meromorphic functions you might be able to get a better answer. In addition, you haven's specified directions, but i've taken it as implicit that you have.
If you have any open subset of C that crosses all 6 relevant integrals, and define an integrable function everywhere except that open region, you can always define the function on that subset such that the complete function satisfies the magic square condition.