The conventional extended Kalman filter(EKF) is proposed under the criterion of minimum mean square error(MMSE) and omits high-order information. This leads to poor estimation of the EKF in non-Gaussian and strongly nonlinear systems. In this paper, for a class of nonlinear non-Gaussian systems in polynomial form, Pseudo-linearizing the system while preserving the high-order information, a new filter is proposed based on the minimum error entropy(MEE) criterion called the minimum error entropy high-order extended Kalman filter (MEE-HEKF). In the MEE-HEKF algorithm, the state estimation problem is transformed into a recursive solution problem in the form of a Kalman filter by a fixed-point iterative approach based on the MEE cost function. The proposed algorithm has better robustness under strong nonlinearity and non-Gaussian noise. Finally, the effectiveness of the new algorithm is illustrated by an example.