Any odd square is of the form $ 8k+1$.Hence we construct the following.let $p$ andf $ q$ be two distinct primes of the form $ 8k+3 $.consider the number $ N=2pq$.we can find infinitely many such numbers since from lagrange we can find infinitely many pairs $p,q.$ obviously $N$ is of the form $8k+2$.this number cannot be the sum of two squares .hence it has to be the sum of at least $10 $ squares.(squares are $ 1 mod 8$).

MMEEvN wrote: this number cannot be the sum of two squares .hence it has to be the sum of at least $10 $ squares.(squares are $ 1 mod 8$). Could you please explain this part?