Let $m\geq 1$ and $k$ be integers.
If $m=p_1^{q_1}...p_l^{q_l}$ is the primary decomposition of $m$, it's easy to see with the Chinese remainder theorem that $k$ is a square mod $m$ iff for all $j\in\{1,...,l\}$ the $k$ is a square mod $p_j^{q_j}$.
Wikipedia states ''A number relatively prime to an odd prime $p$ is a residue modulo any power of $p$ if and only if it is a residue modulo $p$.'' which means for an odd prime $p$, that $k$ is a square mod $p_j^{q_j}$ iff $k$ is a square mod $p_j$. With Euler's criterion, I can then decide if $k$ is a square mod $p$ or not.
To get a full answer to the original question (''when is $k$ a square mod $m$?''), I need to know what happens for the prime $p=2$ and powers of it. Mod $2$, every $k$ is a square. Is there a criterion like Euler's to decide if $k$ is a square mod $2^q$?