How to use Frobenius for finding Square Roots in $GF(2^m)$
No, the relation you have noticed does not work in general, and it is easy to find counterexamples. For example using the two following representations of $\mathbf F_8$.

In $\mathbf F_2[X]/(X^3+X^2+1)$, the square root of (the coset of) $X$ is $X^2+X+1$, whereas in $\mathbf F_2[X]/(X^3+X+1)$ it is $X^2+X$.