In biological swarms, army ants and bees have demonstrated the ability to form functional structures for collaborative tasks. Achieving similar functionality with robot swarms requires forming connections between robots using electrical, magnetic, or mechanical means. Our research introduces the PuzzleBots—robot swarms equipped with passive coupling mechanisms that enable collective behavior. These mechanisms leverage the individual mobility and dexterity of each robot to achieve complex assemblies. By coupling together, PuzzleBots can form both rigid and flexible structures that significantly enhance their ability to navigate challenging terrains. Rigid structures offer high load-bearing and transportation capabilities, while flexible structures provide compliance with environmental geometries. We demonstrated that these assembled structures can be precisely controlled using our distributed Model Predictive Control framework, enabling them to traverse rough and discontinuous terrains. Our results show that passive coupling in robot swarms not only improves terrain traversal capabilities but also enhances adaptability to different environments through reconfiguration, surpassing the capabilities of individual robots.