The proliferation of electronic health records necessitates secure and privacy-preserving data sharing frameworks to combat escalating cybersecurity threats in healthcare. Current systems face critical limitations including centralized data repositories vulnerable to breaches, static consent mechanisms, and inadequate audit capabilities. This paper introduces an integrated blockchain and federated learning framework that enables privacy-preserving collaborative AI across healthcare institutions without centralized data pooling. The proposed approach combines federated distillation for heterogeneous model collaboration with dynamic differential privacy that adapts noise injection to data sensitivity levels. A novel threshold key-sharing protocol ensures decentralized access control, while a dual-layer Quorum blockchain establishes immutable audit trails for all data sharing transactions. Experimental evaluation on clinical datasets (Mortality Prediction and Clinical Deterioration from eICU-CRD) demonstrates that our framework maintains diagnostic accuracy within 3.6% of centralized approaches while reducing communication overhead by 71% and providing formal privacy guarantees. For Clinical Deterioration prediction, the framework achieves 96.9% absolute accuracy on the Clinical Deterioration task with FD-DP at $ϵ$ = 1.0, representing only 0.14% degradation from centralized performance. The solution supports HIPAA-aligned technical safeguards, mitigates inference and membership attacks, and enables secure cross-institutional data sharing with real-time auditability. This work establishes a new paradigm for privacy-preserving healthcare AI that balances data utility, regulatory requirements, and protection against emerging threats in distributed clinical environments.