How to argue depends on what you know about coprime integers.
For example, if you know that $a,b$ being coprime means that there are integers $n,m$ with
$$ an+bm=1, $$
you can argue as follows:
First, if $a^2$ and $b^2$ are coprime, as witnessed by integers $n,m$, so
$$ a^2n+b^2m=1, $$
then $k=an$ and $l=bm$ witness that $a,b$ are coprime:
$$ ak+bl=1. $$
The other direction is more interesting: First, note that if $x,y$ are coprime, so are $x^2,y$:
$$ xn+ym=1\Longrightarrow (xn+ym)^2=1, $$
but $$ (xn+ym)^2= x^2 n^2+y(2xnm+ym^2)= x^2k+yl, $$
with $k=n^2$ and $l=2xnm+ym^2$.
This gives us that if $a,b$ are coprime, $a^2,b$ are coprime. But then, letting $x=b,y=a^2$, this gives us that $x^2=b^2$ and $y=a^2$ are coprime as well, as we wanted.
On the other hand, if you know that being coprime can be verified in terms of common prime divisors, then the argument is simpler, because if $p$ is prime and $p|xy$ then $p|x$ or $p|y$. This means that for any prime $p$, we have that $p|a^2$ iff $p|a$, and the same with $b^2$ and $b$, so the prime divisors of $a^2,b^2$ are the same as those of $a,b$. Hence, one pair is coprime iff the other is coprime.