Is it true that if $P(x)$ is a square number for every integers, then $P(x)=(Q(x))^2$

Question

Let $P(x)$ be a polynomial with integer coefficients and $deg(P(x))>0$. Is it true that if $P(x)$ is a square number for every integers, then $P(X)=(Q(x))^2$, with $Q(x)$ being a polynomial? Is the statement correct if $P(x)$ is a square number for infinite integers, not every integers ?