As advances in technology allow the acquisition of complementary information, it is increasingly common for scientific studies to collect multiple data sets. Large-scale neuroimaging studies often include multiple modalities (e.g., functional MRI, diffusion MRI, and/or structural MRI) and behavioral data, with the aim to understand the relationships between data sets. Classical approaches to data integration utilize transformations that maximize covariance or correlation. In neuroimaging, a popular approach uses principal component analysis for dimension reduction prior to feature extraction via a joint independent component analysis (ICA). We introduce Joint and Individual Non-Gaussian component analysis (JIN) for data integration, where dimension reduction and feature extraction are achieved simultaneously. We focus on information shared in subject score subspaces estimated by maximizing non-Gaussianity, and we also examine information unique to each data set. We apply our method to task activation maps and functional connectivity matrices from the Human Connectome Project, where we discover structure that is related to fluid intelligence.