The precise segmentation of cortical and trabecular bone compartments in high-resolution micro-computed tomography (µCT) scans is crucial for evaluating bone structure and understanding how different medical treatments and mechanical loadings affect bone morphology, offering valuable insights into osteoporosis. In this work, we propose a novel hybrid neural network architecture named Dual-Branch Attention-based Hybrid Network (DBAHNet) for 3D µCT segmentation. DBAHNet combines both transformers and convolution neural networks in a dual-branch fashion and fuses their respective information at each hierarchical level, to better capture long-range dependencies and local features, and for a better understanding of the contextual representation. We train and evaluate DBAHNet on three datasets of high-resolution (<5µm) µCT scans of mouse tibiae. The results show that the proposed DBAHNet achieves state-of-the-art performance by surpassing several popular architectures. Our model also achieves a precise segmentation of the cortical and trabecular bone compartments along different regions of the bone, demonstrating a comprehensive understanding of the bone. Models and code are available at GitHub.