Your first argument, beginning "Let it be initially assumed that A and B have a highest common factor = 1..." does not use the condition that x > 2 and y > 2. Therefore, if your argument were valid, it would follow that A^2 + B^2 = C^n had no solutions with n > 2 odd.
However, 2^2 + 11^2 = 5^3.
This is a counterexample to your purported proof, but not to Beal's conjecture.
