You made an implicit assumption (which happens to not be true) in your argument.
I'll give you a few moments to think about it.... and if you sneak a peak at my other replies, you may spot it.
Yes, I'm assuming that no circle is a square, and therefore a circle is a "not-square", and vice versa. Perhaps it's because I've led a sheltered Euclidian life, but I believe that assumption to be true.