The following result is due to Asplund [A, p.235].
Theorem. If $\varnothing\neq E\subset\mathbb{R}^n$ is closed, and $d(x)=\operatorname{dist}(x,E)$, then $f:\mathbb{R}^n\to\mathbb{R}$ defined by $f(x)=|x|^2-d(x)^2$ is convex.
That means $d^2$ has in some sense the same regularity properties as a convex function and that translates into properties of the distance function. The proof of the above result is very easy, but clever.
We have $$ f(x)=|x|^2-\inf_{y\in E}|x-y|^2=|x|^2+\sup_{y\in E}(-|x-y|^2)=\sup_{y\in E} \big(2\langle x,y\rangle -|y|^2\big). $$ Therefore, $f$ is a supremum of a family of affine functions, and hence it is convex.
[A] Asplund, E.: Cebyvsev sets in Hilbert space. Trans. Amer. Math. Soc. 144 (1969), 235-240.