Prognostics technology is important for the sustainability of solid oxide fuel cell (SOFC) system commercialization, i.e., through failure prevention, reliability assessment, and the remaining useful life (RUL) estimation. To solve SOFC system issues, data-driven prognostics methods based on the dynamic neural network (DNN), one of non-linear models, were investigated in this study. Based on DNN model types, the neural network autoregressive (NNARX) model with external inputs, the neural network autoregressive moving average (NNARMAX) model with external inputs, and the neural network output error (NNOE) were utilized to predict the degradation trend and estimate the RUL. First, the degradation trend prediction was executed to evaluate the correctness of the proposed DNN model structures in the first learning phase. Then, the RUL was estimated on the basis of the degradation trend of the NN models in the second inference phase. The comparison test results show the prediction accuracy of the NNARX model is higher and the RUL estimation can be given within a smaller relative error than the NNARMAX and NNOE models. The evaluation criteria of the root mean square error and mean absolute error of the NNARX model are the smallest among these three models. Therefore, the proposed NNARX model can effectively and precisely provide degradation trend prediction and RUL estimation of the SOFC system.