Okay. I've refreshed my memory of this topic, but I am in no way a professional.
First, I am thinking that your problem statement is incorrect. (Which may mean that I'm incorrect, but anyway...)
For example, take a look at the multiplicative group mod 7: .
Squaring each of these elements gives the quadratic residues by inspection: q = 1, 4. The sum of these is obviously not divisible by 7. I think what you are looking for is the sum of the square of all members of the group. This can be easily shown by looking at the summation:
This turns out to be divisible by 7 as required. (And thus the sum of the residues is also divisible by 7. Prove this!) I leave it to you to look at the general case.
-Dan