For healthcare datasets, it is often impossible to combine data samples from multiple sites due to ethical, privacy, or logistical concerns. Federated learning allows for the utilization of powerful machine learning algorithms without requiring the pooling of data. Healthcare data have many simultaneous challenges, such as highly siloed data, class imbalance, missing data, distribution shifts, and non-standardized variables, that require new methodologies to address. Federated learning adds significant methodological complexity to conventional centralized machine learning, requiring distributed optimization, communication between nodes, aggregation of models, and redistribution of models. In this systematic review, we consider all papers on Scopus published between January 2015 and February 2023 that describe new federated learning methodologies for addressing challenges with healthcare data. We reviewed 89 papers meeting these criteria. Significant systemic issues were identified throughout the literature, compromising many methodologies reviewed. We give detailed recommendations to help improve methodology development for federated learning in healthcare.