Machine learning applications are intensively utilized in various science fields, and increasingly the biomedical and healthcare sector. Applying predictive modeling to biomedical data introduces privacy and security concerns requiring additional protection to prevent accidental disclosure or leakage of sensitive patient information. Significant advancements in secure computing methods have emerged in recent years, however, many of which require substantial computational and/or communication overheads, which might hinder their adoption in biomedical applications. In this work, we propose SecureLR, a novel framework allowing researchers to leverage both the computational and storage capacity of Public Cloud Servers to conduct learning and predictions on biomedical data without compromising data security or efficiency. Our model builds upon homomorphic encryption methodologies with hardware-based security reinforcement through Software Guard Extensions (SGX), and our implementation demonstrates a practical hybrid cryptographic solution to address important concerns in conducting machine learning with public clouds.