  We characterise the integral affine plane curves over a finite field $k$ with
the property that all but finitely many of their $\overline{k}$-points have
coordinates that are $j$-invariants of elliptic curves with isomorphic
endomorphism rings. This settles a finite field variant of the Andr\'e-Oort
conjecture for $Y(1)^2_\mathbb{C}$, which is a theorem of Andr\'e. We use our
result to solve the modular support problem for function fields of positive
characteristic.
