If you take $u$ to be the distance from the boundary of $\Omega$ the two properties you look for are statisfied at every point of differentiability for $u$, indeed in any such point $x$ the gradient of $u$ is given by $x-\pi(x)$ where $\pi(x)$ is the (unique) point realising the distance. Uniqueness follows from the assumption about differentiability and it holds a.e.

If you take $u$ to be the distance from the boundary of $\Omega$ the two properties you look for are statisfied at every point of differentiability for $u$, indeed in any such point $x$ the gradient of $u$ is given by $$\frac{x-\pi(x)}{|x-\pi(x)|}$$ where $\pi(x)$ is the (unique) point realising the distance. Uniqueness follows from the assumption about differentiability and it holds a.e.