This article investigates the composite disturbance filtering (CDF) problem for a class of nonlinear stochastic systems subject to composite disturbances. The concerned disturbances include both the unknown deterministic type and the non-Gaussian stochastic type. In order to obtain the optimal state estimation under the influence of unknown input and non-Gaussian noise, a nonlinear CDF method is developed by resorting to the maximum correntropy criterion (MCC). Faced with the nonlinearity of system model as well as a conditionally linear substructure with respect to unknown input, the marginalized unscented transformation is exploited for computation-efficient statistics propagation, and then the statistical linearization is performed to provide a regression model for simultaneous state and unknown input estimation (SSUIE) under the MCC. The proposed filtering algorithm is demonstrated via a numerical example, and further applied to an integrated navigation system (INS). Simulation results confirm that our method has enhanced disturbance rejection ability and improved estimation accuracy in complex non-Gaussian environments.