Federated learning (FL) strives to enable collaborative training of machine learning models without centrally collecting clients' private data. Different from centralized training, the local datasets across clients in FL are non-independent and identically distributed (non-IID). In addition, the data-owning clients may drop out of the training process arbitrarily. These characteristics will significantly degrade the training performance. This paper proposes a Dropout-Resilient Secure Federated Learning (DReS-FL) framework based on Lagrange coded computing (LCC) to tackle both the non-IID and dropout problems. The key idea is to utilize Lagrange coding to secretly share the private datasets among clients so that each client receives an encoded version of the global dataset, and the local gradient computation over this dataset is unbiased. To correctly decode the gradient at the server, the gradient function has to be a polynomial in a finite field, and thus we construct polynomial integer neural networks (PINNs) to enable our framework. Theoretical analysis shows that DReS-FL is resilient to client dropouts and provides privacy protection for the local datasets. Furthermore, we experimentally demonstrate that DReS-FL consistently leads to significant performance gains over baseline methods.