Let $z_0$ be an essential singularity of $f$,then show that $z_0$ is an essential singularity of $f^2$.
TRY:
If $z_0$ is a removable singularity of $f^2$ then by Riemann's Singularity Theorrem $f^2$ is bounded in a deleted neighbourhood of $z_0\implies f$ is bounded in a deleted neighbourhood of $z_0$ which is false.
Now if $z_0$ is a pole of order $m$ of $f^2\implies f^2(z)=\dfrac{\phi(z)}{(z-z_0)^m}$ where $\phi$ is analytic at $z_0$.
How to show from here that $f$ has a pole of order $m$ at $z_0$?If I show this then only then I can show that $z_0$ is an essential singularity of $f^2$.
Please help.
You are allmost there, show that $z_0$ is neither a removable singularity of $f^2$ and $1/f^2$, you have the argument for $f^2$ it is the same for $1/f^2$.