I apologize for posting another answer, but apparently everyone, including me, missed a very simple proof why the product is never a square, which is indeed suitable for children: just consider everything modulo $4$. If the number ends in $77$ it is congruent to $1$ modulo $4$, and $7 \equiv 3 (\mod 4)$, and $3$ is not a square mod $4$.

I apologize for posting another answer, but apparently everyone, including me, missed a very simple proof why the product is never a square, which is indeed suitable for children: just consider everything modulo $4$. If the number ends in $77$ it is congruent to $1$ modulo $4$, and $7 \equiv 3\pmod 4$, and $3$ is not a square mod $4$.