Hydrogels exhibit remarkable softness and responsiveness to external stimuli, making them attractive for a range of applications from soft robotics to biomedical devices. However, their multiphysics nature, coupled with large deformations and mechanical instabilities, poses significant challenges for computational modeling. To address these issues, we propose a meshless multiphysics framework that leverages enhanced local maximum-entropy (max-ent) interpolants. This approach brings together the adaptability of meshless methods—eliminating the need to remesh under severe geometric distortions—the stability of enhanced local max-ent interpolants, and a monophasic hydrogel theory integrating transient Fickian-type diffusion with quasistatic finite-strain mechanics. We demonstrate the potential of the framework through benchmark tests, including free swelling of various 3D hydrogel geometries and deformation of hollow hydrogel structures in both 2D and 3D. The results show that our method maintains accuracy and robustness for large deformations, suggesting it is well-suited to handle complex shapes encountered in emerging manufacturing techniques like 3D printing.