Here is why. Let $\displaystyle \mathbb{Z}_n,\mathbb{S}_n$ denote respectively the integers mod $\displaystyle n$ and the multiplicative group of nonzero squares mod $\displaystyle n$. We have

$\displaystyle S_n \cong \mathbb{Z}_n/\{\pm 1\}$

and also

$\displaystyle \mathbb{S}_{n} \cong \mathbb{S}_p \times \mathbb{S}_q$.

Therefore $\displaystyle |\mathbb{S}_{n}|=|\mathbb{S}_{p}||\mathbb{S}_{q}|= (p-1)(q-1)/4$. Since we have $\displaystyle \left(\frac{a}{n}\right)=1$ for $\displaystyle (p-1)(q-1)/2 = \phi(pq)/2$ elements $\displaystyle a$ of $\displaystyle \mathbb{Z}_n$, there must be some of those which are *not* elements of $\displaystyle \mathbb{S}_{n}$, i.e. not squares.