Biomaterials that can be reversibly stiffened and shaped could be useful in broad biomedical applications where form-fitting scaffolds are needed. Here we investigate the combination of strong non-linear elasticity in biopolymer networks with the reconfigurability of packed hydrogel particles within a composite biomaterial. By packing microgels into collagen-1 networks and characterizing their linear and non-linear material properties, we empirically determine a scaling relationship that describes the synergistic dependence of the material's linear elastic shear modulus on the concentration of both components. We perform high-strain rheological tests and find that the materials strain stiffen and also exhibit a form of programmability, where no applied stress is required to maintain stiffened states of deformation after large strains are applied. We demonstrate that this non-linear rheological behavior can be used to shape samples that do not spontaneously relax large-scale bends, holding their deformed shapes for days. Detailed analysis of the frequency-dependent rheology reveals an unexpected connection to the rheology of living cells, where models of soft glasses capture their low-frequency behaviors and polymer elasticity models capture their high-frequency behaviors.