In this paper, we present BreastSegNet, a multi-label segmentation algorithm for quantitative analysis of breast MRI images. Unlike previous breast MRI segmentation methods that focus on only a few anatomical structures (e.g., fibrous tissue or tumors), BreastSegNet includes nine anatomical structures: fibrous tissue, blood vessels, muscles, bones, lesions, lymph nodes, heart, liver, and implants. The researchers manually annotated 1,123 MRI slices after review and revision by an expert radiologist. After benchmarking nine segmentation models, including U-Net, SwinUNet, UNet++, SAM, MedSAM, and nnU-Net with multiple ResNet-based encoders, nnU-Net ResEncM achieved the highest performance, achieving an average Dice score of 0.694 for all labels. In particular, it achieved Dice scores exceeding 0.73 for heart, liver, muscle, fibrous tissue, and bone, and approached 0.90 for heart and liver. All model code and weights are publicly available, and data will be made public in the future.