Just choose integers $7$ modulo $8$ and the quadratic residues modulo $8$ are $0, 1, 4$. $7$ cannot be written as sum of $3$ of these numbers.

Modulo $8$, $0^2\equiv4^2\equiv0$, $2^2\equiv6^2\equiv4$, and $1^2\equiv3^2\equiv5^2\equiv7^2\equiv1$, so $0$, $1$, and $4$ are the only quadratic residues. As a result, the positive integers equivalent to $7\pmod{8}$, of which there are infinitely many, cannot be expressed as the sum of three positive perfect squares, as no choice of positive integers $a,b,c\in\{0,1,4\}$, not necessarily distinct, have $a+b+c\equiv7\pmod{8}$.