As governmental health information systems become increasingly digitized, the need for collaborative analytics across decentralized regions has intensified. However, privacy concerns, regulatory constraints, and infrastructure disparities have limited the extent to which sensitive health data can be aggregated and analyzed across jurisdictions. This paper explores the integration of blockchain technology with federated learning (FL) to enable privacy-preserving data analytics across distributed governmental health information systems. By combining FL's decentralized model training capabilities with blockchain’s immutable, transparent ledger and consensus mechanisms, the proposed framework ensures secure, auditable, and policy-compliant data collaboration without requiring raw data exchange. The framework leverages smart contracts to automate access control, consensus validation, and compliance enforcement among participating health institutions. Each node (representing a governmental health entity) trains models locally and shares only encrypted model parameters, which are validated and recorded on the blockchain. This eliminates the need for centralized authorities and reduces the risk of data leakage or manipulation. A core contribution of this work lies in addressing public-sector constraints such as legacy infrastructure, heterogeneous data standards, and institutional trust gaps through a modular, interoperable design. The system includes support for dynamic node participation, real-time updates, and compatibility with health data standards such as HL7 and FHIR. Use-case simulations across municipal, regional, and national health departments demonstrate improved efficiency in outbreak prediction, chronic disease surveillance, and population-level risk stratification while maintaining strict compliance with data protection regulations. This paper advances a scalable and trustworthy architecture for cross-border health collaboration, offering a blueprint for digital public health infrastructures in the age of data sovereignty and distributed intelligence.