But it has nothing to do with n being square -- nothing at all.
It has almost nothing to do with p,q,r,s being prime. The only tie to primes is that, by the Goldbach conjecture (or the Ternary Goldbach Theorem) there are _lots_ of distinct 4-tuples of odd primes whose sum is n. So many 4-tuples that, in general, one of them is sure to satisfy the sum of squares condition.
To understand this point, consider the following question ...
Take any large even positive integer n (with say, n > 10^4, just to be safe). Consider all possible 4-tuples of odd positive integers whose sum is n, and choose them randomly, one at a time, until one of those 4-tuples (a,b,c,d) is such that a^2+b^2+c^2+d^2 is a perfect square. As a function of n, what is the expected number of choices required?
My point is that, for sufficiently large even n, the number of odd prime 4-tuples whose sum is n easily exceeds the expected required number of random odd 4-tuples whose sum is n such that one of those odd 4-tuples satisfies the sum of squares condition.