Combinatorial mechanical metamaterials are engineered structures built up from units that may be combined in many different ways so as to control the overall collective modes and characteristic mechanical response. By using multistable mechanical units, a large design space can be explored depending on whether neighboring units interact in compatible or incompatible ways. Much progress has been made towards avoiding incompatible combinations so as to attain desired collective zero-modes of deformation. Less focus has been given to the possibility of controlled introduction of frustrated interactions intended to direct the propagation of internal stresses. Based on analogy with magnetic artificial spin ice systems, we propose a spring network built up from square, bistable unit cells whose low-energy excitations appear as mesoscopic domain walls between competing ground states. Mechanical incompatibilities become localized within these mesoscopic structures, and they can be manipulated effectively from the boundaries of the system. Our proposed design incorporates mechanical prestress in order to force the square cells to spontaneously adopt one of their two zero-energy deformations. Our system is analogous to square artificial spin ice, for which the low-energy excitations are manifested as topological monopoles and strings. We argue that theoretical descriptions of the low-energy excitations for discrete magnetic degrees of freedom also work to explain the behavior of our mechanical system, even though the underlying degrees of freedom are continuous. Finally, we introduce a more complex prestressed mechanical metamaterial geometry, which is inherently frustrated, degenerate, and exhibits complex defects.