Ah, looking at Mark44's post, there is a nicer way.
First we prove 0*0 = 0.
This is true because 0*0 = 0*(0+0) = 0*0 + 0*0. By uniqueness of the additive identity, 0*0 = 0.
Also, 1 + -1 = 0, by definition.
Thus (1 + -1)*(1 + -1) = 0.
Distribution gives us (1*1 + -1*1) + (1*-1 + -1*-1) = 0...
If you are doing an axiomatic treatment of the integers, you define 1 to be the multiplicative identity. -1 is defined to be the additive inverse of 1 (so 1 + -1 = 0).
-1*-1 = -1*(-1 + 0) [Use that 0 is the additive identity]
-1*(-1 + 0) = -1*(-1 + (1 + -1)) [Use that 0 = 1 + -1]
-1*(-1 + (1 +...