After deriving a general expression for the drag force on an anharmonic oscillator with quartic potential moving through random electromagnetic radiation, we specialize for the thermal and for the zero-point field (ZPF) radiation cases. It is explicitly shown that, as should be physically expected because of ZPF Lorentz invariance, the drag force cancels to all orders in $\ensuremath{\beta}$ and in the oscillator coupling parameter $\ensuremath{\eta}$ for the case of motion through the ZPF. This provides a simple positive test on the self-consistency of the usual techniques of stochastic electrodynamics under nonlinear potentials.