If $F(x,y)=0$ has infinitely many integral solutions, then by Siegel's Theorem the projective closure of this curve has at most $2$ points at infinity. These point are just those with coordinates $(x:y:0)$ with $H(x,y)=0$, where $H=\text{High}(F)$. So the polynomial $H$ has total degree at most $2$ if it is to be separable and not divisible by $x$ nor $y$.

(Added some remarks from my comments)

Suppose that $F=F_1F_2\cdots F_r$ with $F_i$ irreducible. If $F=0$ has infinitely many points of the requested form, then so does one of the factors. Furthermore, $\text{High}(F)=\text{High}(F_1) \text{High}(F_2)\cdots\text{High}(F_r)$. So in order to look at the question, we may assume that $F$ is irreducible. But the $F$ is even absolutely irreducible, which we assume from now on.

As $F(x,y)=0$ has infinitely many integral solutions, then by Siegel's Theorem the projective closure of this curve has at most $2$ points at infinity. These point are just those with coordinates $(x:y:0)$ with $H(x,y)=0$, where $H=\text{High}(F)$. So the polynomial $H$ has total degree at most $2$ if it is to be separable and not divisible by $x$ nor $y$.

So a counterexample (that is where $H(x,y)$ is not squarefree) would have degree at most $2$.

I believe that degree $2$ can be ruled out. However, degree $1$ seems to amount to solve Pillai's conjecture: Given nonzero integers $A,B,C$, then $Au^m+Bv^n=C$ has only finitely many integral solutions $u,v,m,n$ with $m,n\ge3$. You have the additional assumption that $u$ and $v$ are relatively prime. I'm sure that this doesn't make the conjecture easier.