We say that an algebraic variety has the Hilbert Property for integral points if its set of integral points is not thin, which can be seen as a generalization of Hilbert's irreducibility theorem to this variety. A conjecture of Corvaja and Zannier predicts that a smooth simply connected variety with a Zariski dense set of $S$-integral points has the Hilbert Property for $S$-integral points, possibly after a finite enlargement of the base field and the set of places $S$. We will explain how to prove this conjecture for complements of anticanonical divisors in del Pezzo surfaces by using pairs of conic fibrations to produce many integral points. We will also discuss non-potential results for affine cubic surfaces, such as proving that the set of $\mathbb{Z}$-points of $x^3+y^3+z^3=1$ is not thin.