This work is concerned with the relativistic quantum dynamics of a self-interacting neutron in the presence of an external ultra-strong electromagnetic (EM) field in a cylindrical inertial frame. We first regard the Dirac-Pauli (DP) Lagrangian to study the planar dynamics of a neutron polarized along the z-axis subjected to a confining external static EM field composed of a homogeneous magnetic field in the z-direction and a linear radial electric field in the polar plane. The corresponding discrete Landau energy levels are found. As a nonlinear (NL) example model, we introduce a 1-flavor Nambu Jona-Lasinio (NJL) mass term into the DP Lagrangian. The continuous ground-state Landau levels are determined. We readily obtain modified Maxwell's equations associated with these levels. We consider a simple application of the model related to the dynamics of neutrons in the presence of strong-QED fields inside the surface of aligned neutron stars. We briefly comment on possible classical solitonic solutions of the model.