To alleviate wave and vibration transmission in automotive, aerospace, and civil engineering fields, researchers have investigated periodic metamaterials with especially architected internal topologies. Yet, these solutions employ heavy materials and narrowband, resonant phenomena that are unsuitable for the many applications where broadband frequency vibration energy is a concern, such as that injected by impact forces, and weight is a performance penalty. To overcome these limitations, a new idea for lightweight, elastomeric metamaterials constrained near critical points is recently being explored, such that improved shock and vibration damping is achieved using reduced mass than conventional periodic metamaterials. On the other hand, the internal architectures of these metamaterials have not been explored beyond classical circular designs whereas numerous engineering structures involve square or rectangular geometries that may challenge the ability to realize critical point constraints due to the lack of rotational symmetry. The objectives of this research are to undertake a first study of square cross-section elastomeric metamaterials and to assess the impact tolerance of structures into which these metamaterials are embedded and constrained. Finite element simulations guide attention to design parameters for the metamaterial architectures, while experimental efforts quantify the advantages of constraints on enhancing impact tolerance metrics for engineering structures. It is seen that although the architected metamaterial leads to slightly greater instantaneous acceleration amplitude immediately after impact, it more rapidly attenuates the injected energy when compared to the solid and heavier elastomer mass from which the metamaterial is derived.