Hybrid proton exchange membrane fuel cell (PEMFC)-battery-supercapacitor structures have recently been viewed as viable sources of emergency and auxiliary power in all-electric planes since they exhibit high specific energy, fast transient response, and minimal reliance on traditional engine-powered subsystems. Current supervisory energy management approaches towards such systems mainly consider optimization for hydrogen savings or instantaneous efficiency. However, a hydrogen-alone dispatch may lead to higher battery current stress, faster PEMFC dynamic load transitions, increased supercapacitor ripple and larger deviations from steady-state values of the DC-bus voltage. In this paper, a multi-objective health-aware strategy for energy management of a hybrid emergency power system consisting of a PEMFC, a lithium-ion battery and a supercapacitor pack is proposed. The HA-MOSCA supervisory controller incorporates component-level health indices along with system-level efficiency and power quality objectives in the dispatch decision-making process. Hydrogen consumption rate, battery aging metric, fuel cell dynamic stress, supercapacitor stress and DC-bus voltage ripple are minimized concurrently considering power balance, states and operation constraints. A health-aware adaptive sine cosine optimization algorithm is used to solve the problem and obtain close-to-optimum power distribution at low supervisory computational cost. The research is based on an emergency power system architecture for an aircraft comprising a 40 Ah battery, 15.6 F supercapacitor pack, a PEMFC stack and a 270 V DC-bus. Simulation studies performed for comparison against PI control, equivalent consumption minimization strategy and hydrogen-alone sine cosine optimization demonstrate that the proposed strategy reduces battery aging index by 22.8%, fuel cell stress by 28.2%, supercapacitor stress by 21.9% and average DC-bus voltage deviation by 35.9% compared to the hydrogen-only solution, but causes just 1.98% hydrogen loss. When compared against conventional PI control, hydrogen consumption is decreased by 14.4% and average DC-bus voltage deviation is lowered by 56.9%. The findings show that a minor compromise in fuel optimization will yield significant returns in terms of increased reliability, improved electrical performance, and deployability of safety-sensitive auxiliary power units on board airplanes.