HPC and quantum computing are coming together to create a new model of computational power-power that is on one hand speeding up the process of scientific discovery and on the other increasing the security dilemma of data. The chapter examines the convergence of HPC and quantum technologies in the field of healthcare data privacy, where data with sensitive patient information requires the performance and the protection. It starts by analyzing the weakness of the conventional HPC security with a focus on the trade-off between strong encryption and computational efficiency. The research then explores the impending quantum menace: specifically, the so-called harvest-now, decrypt-later attacks, and assesses the post-quantum cryptographic (PQC) and quantum key distribution (QKD) standards as the most important in mitigating the threat. Quantum-Resilient Healthcare HPC (QR-HPC) is a hybrid framework that may be suggested to involve HPC with quantum-safe and privacy-preserving techniques, including homomorphic encryption, federated learning, and trusted execution environments (TEE). The experimental outcomes of a prototype implementation prove that the performance penalty of quantum-safe encryption is relatively small (less than 10%) and provides a high level of protection against classical and quantum-based attacks. These results confirm that computational power and quantum security breakthroughs of HPC can facilitate secure, scalable, and compliant healthcare data analytics, which can make the privacy crossroads a roadmap to sustainable, quantum-resilient computing.