In any structure, the non-definable elements are exactly the elements that realize the type $p(x)$, which is the type containing all assertions of the form:
$$\varphi(x)\to\exists y\neq x\ \varphi(y).$$
The reason is that if $x$ is not definable in a structure $\mathcal{M}$, then it is not the unique satisfying instance of any formula (for this is what it means to be definable), and so if $\varphi(x)$ holds, then there must be some other $y$ also satisfying that formula. And conversely, if $x$ is definable, then there is some formula $\varphi$ for which $x$ is the only satisfying instance of $\varphi(x)$.