Hypergraphs are a common model for multiway relationships in data, and hypergraph semi-supervised learning is the problem of assigning labels to all nodes in a hypergraph, given labels on just a few nodes. Diffusions and label spreading are classical techniques for semi-supervised learning in the graph setting, and there are some standard ways to extend them to hypergraphs.However, these methods are linear models, and do not offer an obvious way of incorporating node features for making predictions.Here, we develop a nonlinear diffusion process on hypergraphs that spreads both features and labels following the hypergraph structure. Even though the process is nonlinear, we show global convergence to a unique limiting point for a broad class of nonlinearities and we show that such limit is the global minimum of a new regularized semi-supervised learning loss function which aims at reducing a generalized form of variance of the nodes across the hyperedges.The limiting point serves as a node embedding from which we make predictions with a linear model. Our approach is competitive with state-of-the-art graph and hypergraph neural networks, and also takes less time to train.