Longitudinal studies often entail categorical outcomes as primary responses. When dropout occurs, non-ignorability is frequently accounted for through shared parameter models (SPMs). In this context, several extensions from Gaussian to non-Gaussian longitudinal processes have been proposed. In this paper, we formulate an approach for non-Gaussian longitudinal outcomes in the framework of joint models. As an extension of SPMs, based on shared latent effects, we assume that the history of the response up to current time may have an influence on the risk of dropout. This history is represented by the current, expected, value of the response. Since the time a subject spends in the study is continuous, we parametrize the dropout process through a proportional hazard model. The resulting model is referred to as Generalized Linear Mixed Joint Model (GLMJM). To estimate model parameters, we adopt a maximum likelihood approach via the EM algorithm. In this context, the maximization of the observed data log-likelihood requires numerical integration over the random effect posterior distribution, which is usually not straightforward; under the assumption of Gaussian random effects, we compare Gauss-Hermite and Pseudo-Adaptive Gaussian quadrature rules. We investigate in a simulation study the behaviour of parameter estimates in the case of Poisson and Binomial longitudinal responses, and apply the GLMJM to a benchmark dataset.