Integrating 2D mammography with 3D magnetic resonance imaging (MRI) is crucial for improving breast cancer diagnosis and treatment planning. However, this integration is challenging due to differences in imaging modalities and the need for precise tissue segmentation and alignment. This paper addresses these challenges by enhancing biomechanical breast models in two main aspects: improving tissue identification using nnU-Net segmentation models and evaluating finite element (FE) biomechanical solvers, specifically comparing NiftySim and FEBio. We performed a detailed six-class segmentation of breast MRI data using the nnU-Net architecture, achieving Dice Coefficients of 0.94 for fat, 0.88 for glandular tissue, and 0.87 for pectoral muscle. The overall foreground segmentation reached a mean Dice Coefficient of 0.83 through an ensemble of 2D and 3D U-Net configurations, providing a solid foundation for 3D reconstruction and biomechanical modeling. The segmented data was then used to generate detailed 3D meshes and develop biomechanical models using NiftySim and FEBio, which simulate breast tissue's physical behaviors under compression. Our results include a comparison between NiftySim and FEBio, providing insights into the accuracy and reliability of these simulations in studying breast tissue responses under compression. The findings of this study have the potential to improve the integration of 2D and 3D imaging modalities, thereby enhancing diagnostic accuracy and treatment planning for breast cancer.