The purpose of this study is to determine the material parameters of platinum-cured silicone using uniaxial tensile testing combined with hyperelastic material models to simulate material deformation via the finite element method (FEM). Experimental data from uniaxial tensile tests were used to estimate and capture the material behavior based on selected hyperelastic models. Matlab software was employed to fit experimental data and calculate material parameters, which were subsequently implemented in FEM simulations using Ansys Workbench. Simulation results were compared with experimental data. The findings indicate that the Mooney–Rivlin model provides the best agreement with the experimental results. Initial numerical simulations, assuming ideal incompressibility, produced reasonable approximations but failed to fully reproduce actual behavior. Therefore, material compressibility must be considered. By introducing near-incompressibility coefficients corresponding to Poisson’s ratios, it was found that a value of 0.41 yielded very high consistency between experimental and simulated results. Modeling the nonlinear elastic deformation of hyperelastic materials is critical for both material development and product design. This study highlights the importance of incorporating compressibility in modeling, thereby improving accuracy and practical applicability. The outcomes provide a foundation for employing platinum-cured silicone simulations in the design and fabrication of external breast prostheses for post-mastectomy patients.