This work introduces a novel approach to model regularization and explanation in Vision Transformers (ViTs), particularly beneficial for small-scale but high-dimensional data regimes, such as in healthcare. We introduce stochastic embedded feature selection in the context of echocardiography video analysis, specifically focusing on the EchoNet-Dynamic dataset for the prediction of Left Ventricular Ejection Fraction (LVEF). Our proposed method, termed Gumbel Video Vision-Transformers (G-ViTs), augments Video Vision-Transformers (V-ViTs), a performant transformer architecture for videos with Concrete Autoencoders (CAEs), a common dataset-level feature selection technique, to enhance V-ViT's generalization and interpretability. The key contribution lies in the incorporation of stochastic token selection individually for each video frame during training. Such token selection regularizes the training of V-ViT, improves its interpretability, and is achieved by differentiable sampling of categoricals using the Gumbel-Softmax distribution. Our experiments on EchoNet-Dynamic demonstrate a consistent and notable regularization effect. The G-ViT model outperforms both a random selection baseline and standard V-ViT. The G-ViT is also compared against recent works on EchoNet-Dynamic where it exhibits state-of-the-art performance among end-to-end learned methods. Finally, we explore model explainability by visualizing selected patches, providing insights into how the G-ViT utilizes regions known to be crucial for LVEF prediction for humans. This proposed approach, therefore, extends beyond regularization, offering enhanced interpretability for ViTs.