I believe the knowledge that Z_p\{0} is a group under multiplication modulo p is enough:

|Z_p\{0}| = p-1 = 2 (mod 4) thanks to the condition: p = 3 (mod 4).
By Lagrange’s theorem the order of each element of a group must divide the order of the group, therefore Z_p\{0} does not contain an element of order 4. As such, if x^4=1, we must have either x^2=1 or x=1, in both cases the result follows.