This paper proposes an event-driven model predictive control scheme with guaranteed closed-loop convergence properties for the class of integral continuous-time hybrid automata (icHA). After converting icHA to a corresponding event-driven representation that allows one to compute the model predictive control action by mixed integer programming, sufficient conditions ensuring event-asymptotic and time-asymptotic convergence are proven. The paper also shows how the same modeling methodology can be employed to efficiently solve problems of verification of safety properties.