At least one of the three can definitely be a square; just take any triple and scale it by a number that makes one of the three a square, like 9, 12, 15. As for proving that there can only be one square, the generating equations
a = n^2 - m^2
b = 2nm
c = n^2 + m^2
seem promising. I believe these equations generate all pythagorean triples for non-negative integer values of n and m.