Accurate simulations are essential for engineering applications, and intricate continuum mechanical material models are constructed to achieve this goal. However, the increasing complexity of the material models and geometrical properties leads to a significant increase in computational effort. Model order reduction aims to implement efficient methods for accelerating the simulation process while preserving a high degree of accuracy. In the present work, hyper-reduced order modeling of structural simulations with a two-surface gradient-extended damage-plasticity model are investigated. In these simulations, path-following methods have to be used in order to compute past a certain point, where snapback behavior occurs. The proper orthogonal decomposition-based discrete empirical interpolation method is introduced and explained in the context of quasi-static nonlinear solid mechanics. To show the applicability of the methodology, a numerical example is investigated. The results show that a reduced order model can be created that has a high accuracy and speeds up the simulation significantly in case of brittle damage. For damage-plasticity, the combination of the path-following method and model order reduction can lead to accurate and fast simulations but also simulation instabilities in certain cases.