A stochastic system with double singularities driven by non-Gaussian noise is investigated. The Fokker-Plank equation of the system is obtained through the path-integral approach and the method of transformation. Based on the definition of Shannon’s information entropy, the time dependence of entropy flux and entropy production of the system is calculated both in the absence and in the presence of non-equilibrium constraint. The influences of the system dissipative parameter, the system singularity strength parameter, the noise correlation time and the noise deviation parameter on entropy flux and entropy production are discussed.