Let $X $ be a completely regular topological space and let $U $ be an open subset of $X $. I am looking for an equivalent condition on $U $ such that if $C_1$ and $C_2$ are two closed subsets of $X $ with $U\subseteq C_1\cup C_2$, then $U\subseteq C_1$ or $U\subseteq C_2$.
For example, isolated points have that property.
Very generally, a topological space with this property (that it cannot be written as a union of two proper closed sets) is called irreducible. So, what you are asking for is for $U$ to be irreducible as a subspace of $X$. A Hausdorff space with more than one point is never irreducible: pick distinct points $x,y$ and disjoint open sets containing them, and then the complements of these open sets will be closed proper subsets whose union is the whole space.
So, if your definition of "completely regular" includes that $X$ is $T_0$ (and thus Hausdorff), the only such open subsets $U$ are the empty set and singletons. (If $X$ is not required to be $T_0$, its $T_0$ quotient will still be Hausdorff, and so the image of $U$ in the $T_0$ quotient has at most one point; equivalently, $U$ must have the indiscrete topology.)