Depending on what Greg's intentions were, the answer may actually be yes.
The current definition of $\overline X$ in the question is $\mathrm{Spec} \mathscr O_X$ which, if intentional, suggests that the global sections of the structure sheaf over $X$ is the same as over $\overline X$. If this is the case, then a principal open set does not qualify.
So, assuming this is the case, the answer is still maybe.
Let's assume that the natural embedding induces an isomorphism $\Gamma(X,\mathscr O_X)\simeq \Gamma(\overline X,\mathscr O_{\overline X})$. Then if $X$ is normal, or at least $S_2$, then this means that the complement of $X$ has at least codimension $2$ and then the same condition holds for every reflexive sheaf.