My proof is below. I am not sure about the second part.
Proof:
First I prove $|x|<|y| \implies x^2<y^2$.
Suppose $|x|<|y|$. Then,
\begin{align*}
|x||x| < |y||x| &\implies x^2<|y||x|\tag1\\
|y||x| < |y||y| &\implies |y||x| < y^2\tag2
\end{align*}
By $(1)$ and $(2)$, $x^2<y^2$
Next I prove $x^2<y^2 \implies |x|<|y|$. For that consider the contrapositive.
$$|x| \geq |y| \implies x^2 \geq y^2$$
From previous case $|x|>|y| \implies x^2>y^2$ is true. Therefore the contrapositive is true. Hence the result follows.
Is my proof correct ?