This study explores strategies to enhance the resilience of electricity supply within microgrids by integrating hydrogen storage systems. Given the inherent complexity of the electrochemical models governing electrolyzers and fuel cells, conventional linear approximations prove inadequate for scheduling. To address this challenge, a novel methodology is proposed, leveraging a piecewise linear model approximation enhanced with advanced feasibility projection techniques.The effectiveness of the proposed optimization framework is comprehensively evaluated using a range of resilience performance metrics, including loss- of-load, duration-of-outage, and system cost. Through extensive simulations and meticulous analysis, the study demonstrates significant improvements in resilience performance. Notably, a substantial reduction in duration-of-outage (13% to 48%) and system cost (6.4% to 21.7%) is achieved, along with a 95% reduction in loss-of-load for critical loads compared to conventional linear model approaches. Furthermore, the proposed optimization framework exhibits a remarkable level of consistency with benchmarks established using accurate nonlinear electrochemical models, typically staying below 1% across various evaluation metrics. These findings underscore the effectiveness and practical feasibility of the proposed methodology in enhancing the resilience of microgrid electrical power systems.