Automated segmentation of BUS images is important for precise lesion delineation and tumor characterization, but is challenged by inherent artifacts and dataset inconsistencies. In this work, we evaluate the use of a modified Residual Encoder U-Net for breast ultrasound segmentation, with a focus on uncertainty quantification. We identify and correct for data duplication in the BUSI dataset, and use a deduplicated subset for more reliable estimates of generalization performance. Epistemic uncertainty is quantified using Monte Carlo dropout, deep ensembles, and their combination. Models are benchmarked on both in-distribution and out-of-distribution datasets to demonstrate how they generalize to unseen cross-domain data. Our approach achieves state-of-the-art segmentation accuracy on the Breast-Lesion-USG dataset with in-distribution validation, and provides calibrated uncertainty estimates that effectively signal regions of low model confidence. Performance declines and increased uncertainty observed in out-of-distribution evaluation highlight the persistent challenge of domain shift in medical imaging, and the importance of integrated uncertainty modeling for trustworthy clinical deployment. \footnote{Code available at: https://github.com/toufiqmusah/nn-uncertainty.git}