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

Is there $f$ for which:

1 Both $x,y$ are squares infinitely often, i.e. $x=u^2,y=v^2$?

2 Both $x,y$ are (arbitrary) large powers infinitely often: $x=u^k,y=v^m$ for some $k,m \ge 2$ (the larger the better)?

For large $k,m$ the n-conjecture implies the number of monomials can't be too small.