As health information crosses from one healthcare domain or infrastructure to another, robust confidentiality, authentication, and integrity of the electronic medical records are crucial. This study aims to present a hybrid post-quantum encryption and authentication scheme for secure medical data exchange across the border using lattice-based encryption. The framework integrates a session-key establishment mechanism based on ML-KEM, AES-256-GCM for medical-data encryption, a digital signature scheme based on ML-DSA, integrity check based on hash function and identity-bound transaction metadata. It takes into account the mathematical security model involving key encapsulation, signature verification, adversarial advantage, communication overhead and processing cost. This shows that the proposed framework takes around 0.47ms for key encapsulation, 0.55ms for decapsulation and 0.94ms for digital signing, using simulation-based evaluation. The assumed end-to-end processing time is 83.91ms for a medical record of 50 MB, and the security related communication overhead falls off to about 0.011% when the size of the medical record increases. The results of the analysis show that quantum-resistant protection can be implemented with a moderate computational and communication overhead in cross-border healthcare settings.