Using Church's $\lambda x.(\lambda y.y))$ as false and $\lambda x.(\lambda y.x))$ as true, and given two free variables $g$ and $h$:
Could there exist a function $eq?$ such that $(eq?\ g\ h)$ is false, and $(eq?\ g\ g)$ is true?
I typically would show my work here, but as of right now, I honestly have nothing except a hunch that it's categorically impossible.