Accurate assessment of human epidermal growth factor receptor 2 (HER2) expression is critical for breast cancer diagnosis, prognosis, and therapy selection; yet, most existing digital HER2 scoring methods rely on bulky and expensive optical systems. Here, we present a compact and cost-effective lensfree holography platform integrated with deep learning for automated HER2 scoring of immunohistochemically stained breast tissue sections. The system captures lensfree diffraction patterns of stained HER2 tissue sections under RGB laser illumination and acquires complex field information over a sample area of ~1,250 mm^2 at an effective throughput of ~84 mm^2 per minute. To enhance diagnostic reliability, we incorporated an uncertainty quantification strategy based on Bayesian Monte Carlo dropout, which provides autonomous uncertainty estimates for each prediction and supports reliable, robust HER2 scoring, with an overall correction rate of 30.4%. Using a blinded test set of 412 unique tissue samples, our approach achieved a testing accuracy of 84.9% for 4-class (0, 1+, 2+, 3+) HER2 classification and 94.8% for binary (0/1+ vs. 2+/3+) HER2 scoring with uncertainty quantification. Overall, this lensfree holography approach provides a practical pathway toward portable, high-throughput, and cost-effective HER2 scoring, particularly suited for resource-limited settings, where traditional digital pathology infrastructure is unavailable.