In Conway's book The Sensual (Quadratic) Form he covers Zolotarev's proof of quadratic reciprocity:

The legendre symbol (a|m) is defined as the sign of the permutation "multiplication by a mod m". This happens to match up with the usual definition. (Note the Cayley type replacement of 'a' with the function 'multiplication by a').

Then quadratic reciprocity is proved just using group theory, and as Conway points out this has no mention of square number or even prime numbers!