A polynomial $f(x)$ is a factor-square polynomial if $f(x)|f(x^2)$. For example, $x-1|x^2-1$, so $x-1$ is a factor-square polynomial.
Here is a list of polynomials degrees 1 and 2: $$, $−1$; $x^2$, $x^2-1$, $x^2-x$, $x^2+x+1$, $x^2-2x+1$.
After generating some more of these types of polynomials, it appears that either all of the coefficients are real, or all complex and irreducible over reals.
Are there monic factor-square polynomials of some degree with real number coefficients, but some of those coefficients are not integers?
First notice that if $\alpha$ is a root of $f$, then $\alpha^2$ is also a root of $f$ and, since $f$ has a finite number of roots, we should have $\alpha^{2^i}=\alpha^{2^j}$, thus $\alpha=0$ or it is a root of unity. This helps us narrow the search. For an example, let $\omega$ be a primitive root of unity with $\omega^{2^n}=1$. Then consider $$f(X)=(X-1)^2\prod_{i=1}^{n-1} (X-\omega^{2^i})(X-\overline{\omega}^{2^i}).$$
This obviously has real coefficients, but it will not have integer coefficients. Assuming the contrary, note that it has $\omega^2$ as a root which is a primitive root of order $2^{n-1}$. Since the cyclotomic polynomial $\Phi_{2^{n-1}}(X)$ is the minimal polynomial of $\omega^2$ over $\mathbb{Z}[X]$, we should have $\Phi_{2^{n-1}}(X)(X-1)^2|f(x)$, thus looking at degrees $2^{n-2}=\varphi(2^{n-1})\leq 2n-2$, which is false once $n\geq 6$.