Let $f(x,y)=0$ be (irreducible) elliptic curve over the rationals.

Is it possible:

1 Both $x,y$ to be squares infinitely often?

2 Both $x,y$ to be (arbitrary) large powers, not necessarily the same power?

For Weierstrass model, $x$ square infinitely often is possible, though I believe abc implies both can't be squares infinitely often.

For larger powers the n-conjecture implies the number of monomials can't be too small.