Consider going a small distance $h$ along the diagonal away from a corner. The probability of finding a point in that corner is $n h^2$, and such a point is at least $x = \sqrt{2} - 2h$ away from a corresponding point in the opposite corner. If $D$ is the maximum distance, we have for small enough $h$ $$\Pr[D > \sqrt{2} - 2h] \approx n h^2 .$$
Rewriting $h = (\sqrt{2}-x)/2$, we hence get the approximate distribution function $$F(x) = \Pr[D \le x] \approx 1 - \frac{n}{4} (\sqrt{2} - x)^2$$ with the lowest acceptable $x_0$ where $F(x_0) = 0$ or $x_0 = \sqrt{2} - 2/\sqrt{n}$. Integrating then gives the expectation, $$E[D] \approx \int_{x_0}^{\sqrt{2}} \! x \, F'(x) \, dx = \sqrt{2} - \frac{4}{3\sqrt{n}}.$$