The amount of biomedical data continues to grow rapidly. However, collecting data from multiple sites for joint analysis remains challenging due to security, privacy, and regulatory concerns. To overcome this challenge, we use Federated Learning, which enables distributed training of neural network models over multiple data sources without sharing data. Each site trains the neural network over its private data for some time, then shares the neural network parameters (i.e., weights, gradients) with a Federation Controller, which in turn aggregates the local models, sends the resulting community model back to each site, and the process repeats. Our Federated Learning architecture, MetisFL, provides strong security and privacy. First, sample data never leaves a site. Second, neural network parameters are encrypted before transmission and the global neural model is computed under fully-homomorphic encryption. Finally, we use information-theoretic methods to limit information leakage from the neural model to prevent a curious site from performing model inversion or membership attacks. We present a thorough evaluation of the performance of secure, private federated learning in neuroimaging tasks, including for predicting Alzheimer's disease and estimating BrainAGE from magnetic resonance imaging (MRI) studies, in challenging, heterogeneous federated environments where sites have different amounts of data and statistical distributions.