The strict security requirements placed on medical records by various privacy regulations become major obstacles in the age of big data. To ensure efficient machine learning as a service schemes while protecting data confidentiality, in this work, we propose blind UNET (BUNET), a secure protocol that implements privacy-preserving medical image segmentation based on the UNET architecture. In BUNET, we efficiently utilize cryptographic primitives such as homomorphic encryption and garbled circuits (GC) to design a complete secure protocol for the UNET neural architecture. In addition, we perform extensive architectural search in reducing the computational bottleneck of GC-based secure activation protocols with high-dimensional input data. In the experiment, we thoroughly examine the parameter space of our protocol, and show that we can achieve up to 14x inference time reduction compared to the-state-of-the-art secure inference technique on a baseline architecture with negligible accuracy degradation.