We characterize the model companions of universal Horn classes generated by a two-element algebra (or ordered two-element algebra). We begin by proving that given two mutually model consistent classes $\mathbf{M}$ and $\mathbf{N}$ of $\mathscr{L}$ (respectively $\mathscr{L}'$) structures, with $\mathscr{L} \subseteq \mathscr{L}'$, $\mathbf{M}^{\mathrm{ec}} = \mathbf{N}^{\mathrm{ec}}\mid_\mathscr{L}$, provided that an $\mathscr{L}$-definability condition for the function and relation symbols of $\mathscr{L}'$ holds. We use this, together with Post's characterization of $\mathbf{ISP}(A)$, where $A$ is a two-element algebra, to show that the model companions of these classes essentially lie in the classes of posets and semilattices, or characteristic two groups and relatively complemented distributive lattices.