= Solution
Since a square is nonnegative, $F_Y(y)=0$ for $y<0$. For $y\geq0$, the event $X^2\leq y$ is $-\sqrt y\leq X\leq\sqrt y$. As $X$ has no point masses,
$$
\boxed{F_Y(y)=\begin{cases}
0,&y<0,\\
F_X(\sqrt y)-F_X(-\sqrt y),&y\geq0.
\end{cases}}
$$
Differentiating for $y>0$ gives the contributions of the two inverse branches:
$$
\boxed{f_Y(y)=\begin{cases}
\dfrac{f_X(\sqrt y)+f_X(-\sqrt y)}{2\sqrt y},&y>0,\\
0,&y<0.
\end{cases}}
$$
The value assigned to the <probability density function> at $y=0$ is arbitrary. It can be unbounded near $0$, but there is no atom there because $\mathbb P(X=0)=0$. Both positive and negative preimages must be included; keeping just one would generally lose probability.
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