As healthcare systems transition into an era dominated by quantum technologies, the need to fortify cybersecurity measures to protect sensitive medical data becomes increasingly imperative. This paper navigates the intricate landscape of post-quantum cryptographic approaches and emerging threats specific to the healthcare sector. Delving into encryption protocols such as lattice-based, code-based, hash-based, and multivariate polynomial cryptography, the paper addresses challenges in adoption and compatibility within healthcare systems. The exploration of potential threats posed by quantum attacks and vulnerabilities in existing encryption standards underscores the urgency of a change in basic assumptions in healthcare data security. The paper provides a detailed roadmap for implementing post-quantum cybersecurity solutions, considering the unique challenges faced by healthcare organizations, including integration issues, budget constraints, and the need for specialized training. Finally, the abstract concludes with an emphasis on the importance of timely adoption of post-quantum strategies to ensure the resilience of healthcare data in the face of evolving threats. This roadmap not only offers practical insights into securing medical data but also serves as a guide for future directions in the dynamic landscape of post-quantum healthcare cybersecurity.