No, but there are a few special cases. Take an arbitrary convex function such as $y=x^2$, and take as the domain of $u$ all the points with $y<x^2$. Then for each point of the domain, there is a unique closest point on the graph. Take $u$ to be the distance to that point. The simplest case is $y=0$, $u(x,y) = |y|$. The level
sets of $u$ are curves ``parallel'' to the boundary, and even for $y=x^2$ you can't write a nice formula for $u$, though you can write an implicit one. Same idea works in any number of dimensions, take a convex set, and define $u(x)$ as the distance to the set, when $x$ is exterior to the set. This solves the eikonal equation because $|\nabla u|$ is the maximum rate of change of $u$, which occurs by moving $x$ directly away from that closest point, and when you do that move, $u$ changes by exactly the same distance that you moved $x$.