Arrays of coupled phonon cavities each including an impurity qubit in silicon are considered. We study experimentally feasible architectures that can exhibit quantum many-body phase transitions of phonons, e.g., Mott insulator and superfluid states, due to a strong phonon-phonon interaction (which is mediated by the impurity qubit-cavity phonon coupling). We investigate closed equilibrium systems as well as driven dissipative nonequilibrium systems at zero and nonzero temperatures. Our results indicate that quantum many-body phonon systems are achievable both in on-chip nanomechanical systems in silicon and distributed Bragg reflector phonon cavity heterostructures in silicon-germanium. Temperature and driving field are shown to play a critical role in achieving these phonon superfluid and insulator states, results that are also applicable to polariton systems. Experimental procedures to detect these states are also given. Cavity-phoniton systems enable strong phonon-phonon interactions as well as offering long wavelengths for forming extended quantum states; they may have some advantage in forming truly quantum many-body mechanical states as compared to other optomechanical systems.