  We investigate the Casimir effect between two-dimensional electron systems
driven to the quantum Hall regime by a strong perpendicular magnetic field. In
the large separation (d) limit where retardation effects are essential we find
i) that the Casimir force is quantized in units of 3\hbar c \alpha^2/(8\pi^2
d^4), and ii) that the force is repulsive for mirrors with same type of
carrier, and attractive for mirrors with opposite types of carrier. The sign of
the Casimir force is therefore electrically tunable in ambipolar materials like
graphene. The Casimir force is suppressed when one mirror is a charge-neutral
graphene system in a filling factor \nu=0 quantum Hall state.
