The noise-induced escape problem in stochastic systems has appeared in many fields. In most of the existing theoretical studies on this topic, the mean first-passage time (MFPT) is introduced to quantify the characteristic quantity of escape process. It has been applied to many problems, such as the switching time problem of electronic devices and the life of bistable devices problem. In this paper, we study the MFPT in a piecewise nonlinear model driven by correlated noises for the cases of multiplicative non-Gaussian noise and an additive Gaussian white noise. By using the path integral method, the unified colored noise approximation and the steepest-descent approximation method, the expression of the MFPT is derived. Numerical computation results show that, under the effect of non-Gaussian noise deviating from the Gaussian distribution, correlation time of the non-Gaussian noise and cross-correlation strength, increasing the intensity of non-Gaussian noise leads to the appearance of a one-peak structure in the MFPT. However, the MFPT of the system decreases with the increase of the additive white noise intensity. This shows that MFPT corresponding to the non-Gaussian noise and Gaussian white noise exhibits very different behavior. In addition, we also discuss the effect of the degree of non-Gaussian noise deviating from the Gaussian distribution, correlation time of the non-Gaussian noise and cross-correlation strength on MFPT.