We study the John term of Fab Four cosmology in the presence of a scalar potential. We show here how this theory can describe a wide range of cosmological solutions. This theory has two general functions of the scalar field: the potential $V\left(\varphi \right)$ and the John coefficient function $V_j\left(\varphi \right)$. We show that for very simple choices of those functions, we can describe an accelerated expansion, a radiation-dominated era, and a matter-dominated era. By means of simple modifications, it is also possible to describe nonsingular bouncing versions of those solutions and cyclic universes. We also address some quantum issues of that theory, showing that, for the most significant singular cases, the theory admits a classically well behaved quantization, even though the Hamiltonian has fractional powers in the momenta.