Suppose there are integers x,y,z such that x^4 - y^4 = z^2.  This can be written as a Pythagorean triple (y^2)^2 + z^2 = (x^2)^2. If z is even this implies  y^2 = p^2 - q^2,  z = 2pq, and x^2 = p^2 + q^2, where x and y are both odd, from which we have p^4 - q^4 = (xy)^2.  Therefore, the existence of a solution with even z implies the existence of a solution of the original equation with odd z, so we need only prove that a solution with odd z is impossible. Assuming odd z, the Pythagorean triple implies y^2 = 2pq,  z = p^2 - q^2, and x^2 = p^2 + q^2.  Since 2pq is a square, we can set q = 2u^2 and p = v^2.  Also, from the Pythagorean triple x^2 = p^2 + q^2 we have  p = r^2 - s^2, q = 2rs, and x = r^2 + s^2. Now, since 2u^2 = 2rs, it follows that r = g^2 and s = h^2.  These, along with p = v^2, can be substituted back into p = r^2 - s^2 to give v^2 = g^4 - h^4, where v is smaller than z, contradicting the fact that there must be a smallest solution.