Quadratic points on double planes

Zariski dense collections of quadratic points on curves $X$ are well-understood by results of Harris--Silverman and Vojta, but when $\dim X \geq 2$ there is not an analogous geometric characterization, even conjecturally. In this note we consider the case of a double cover $\pi \colon X \to \mathbb{P}^r$, where Hilbert's Irreducibility Theorem implies that the quadratic points in the fibers of $\pi$ are dense. We show that Vojta's Conjecture implies that, once the canonical bundle of $X$ is sufficiently positive, there are no other sources of Zariski dense quadratic points. This is complemented by several examples of surfaces $X \to \mathbb{P}^2$ with an additional source of dense quadratic points.