Suppose that $F$ is a finite field and $R=M_n(F)$ is the ring of $n$-square matrices over $F$. Here we characterize when the Cayley graph of the additive group of $R$ with respect to the set of invertible elements of $R$, called the unitary Cayley graph of $R$, is well-covered. Then we apply this to characterize all finite rings with identity whose unitary Cayley graph is well-covered or Cohen-Macaulay.