Lung ultrasound (LUS) is a non-invasive bedside imaging technique for diagnosing pulmonary conditions, especially in critical care settings. A-lines and B-lines are important features in LUS images that help to assess lung health and identify changes in lung tissue. However, accurately detecting and segmenting these lines remains challenging, due to their subtle blurred boundaries. To address this, we propose TransBound-UNet, a novel segmentation model that integrates a transformer-based encoder with boundary-aware Dice loss to enhance medical image segmentation. This loss function incorporates boundary-specific penalties into a hybrid Dice-BCE formulation, allowing for more accurate segmentation of critical structures. The proposed framework was tested on a dataset of 4599 LUS images. The model achieved a Dice Score of 0.80, outperforming state-of-the-art segmentation networks. Additionally, it demonstrated superior performance in Specificity (0.97) and Precision (0.85), with a significantly reduced Hausdorff Distance of 15.13, indicating improved boundary delineation and overall segmentation quality. Post-processing techniques were applied to automatically detect and count A-lines and B-lines, demonstrating the potential of the segmented outputs in diagnostic workflows. This framework provides an efficient solution for automated LUS interpretation, with improved boundary precision.