Many disciplines produce count data that contain many zeros. Zero inflation count models such as ZIP and ZIB have been widely used to model count data, in particular, to model the latent structure in a ZIP regression model that allows a non-linear functional relationship between covariates and the expected count outcome. A critical problem in modeling the count response data is the appropriate choice of links functions. Commonly used link functions such as logit link have fixed skewness but lack in flexibility to allow the data to determine the degree of the skewness. To overcome this limitation, we propose a flexible ZIP regression model that combines a generalized extreme value link function with the other link functions. The maximum likelihood estimator is used in the estimation of the model. Maximum likelihood estimation is effective in this model in a series of scenarios. Through simulated data sets and analysis of the healthcare utilization, we show that the proposed link function is quite flexible and performs better against link misspecification than standard link functions.