Abstract: Silicone rubber foam is of significant application value in extreme-environment protection for aerospace, nuclear shielding, and special functional structures. As a lightweight porous polymer with a semi-inorganic Si–O–Si backbone, it exhibits excellent thermal stability, energy absorption, and buffering capacity. However, microscopic damage induced by pulsed irradiation can substantially alter its macroscopic compressive mechanical behavior, thereby affecting the structural integrity and functional reliability of protective systems under extreme service conditions. To investigate how different types of pulsed irradiation affect the quasi-static compressive response of silicone foam, this study conducted systematic uniaxial compression experiments on specimens subjected to pulsed neutron–γ combined irradiation and pulsed X-ray irradiation. The irradiation experiments were performed on a pulsed irradiation facility with high dose-rate characteristics, and three irradiation fluence levels were designed to ensure comparability among different working conditions. Quasi-static compression tests were subsequently carried out on a universal material testing machine under displacement control, and the engineering stress–strain curves were obtained. The effects of irradiation type were quantified through parametric analysis based on a compressible hyper-elastic constitutive model. The results indicate that all irradiated specimens retained the characteristic strongly nonlinear compressive response of silicone foam, encompassing an initial elastic stage, a plateau stage, and a densification stage. Nevertheless, marked differences were observed in elastic modulus and densification onset among the different irradiation groups. Specifically, pulsed neutron–γ combined irradiation led to a pronounced decrease in elastic modulus together with a significant increase in plateau stress, accompanied by an elevated densification strain, suggesting a competitive mechanism between secondary cross-linking and displacement damage. In contrast, pulsed X-ray irradiation produced negligible changes in compressive properties within the experimental dose range, attributable to the strong radiation resistance of phenyl silicone foam. Monte Carlo simulations were further employed to analyze the energy deposition characteristics, spatial distribution, and transient thermal responses associated with each irradiation type, thereby elucidating the underlying damage mechanisms. The experimental stress–strain curves were accurately described by a second-order Ogden–Hyperfoam model within the 0–60% strain range, providing a robust constitutive basis for finite element simulations of pulsed irradiation damage in silicone foam. These findings offer valuable guidance for the mechanical characterization, constitutive modeling, and structural response prediction of silicone foam materials under pulsed irradiation environments.