Recent theoretical and experimental work has demonstrated that acoustic wave tunneling and energy squeezing can be achieved using density- near-zero (DNZ) metamaterial channels [Fleury et al, J. Acoust. Soc. Am., 132(3), 1956 (2012)]. These channels are directly analogous to supercoupling of electromagnetic waves in near-zero permittivity channels. In optics, the field enhancement and uniformity of response within a near-zero permittivity channel can be employed to produce switching behavior, harmonic generation, and wave mixing even with low amplitude input intensities. These optical channels have been already shown to significantly outperform enhancement of nonlinearity in conventional Fabry-Pérot resonant gratings [C. Argyropoulos, et al., Phys. Rev. B, 85, 045129 (2012)]. The analogous properties of velocity field within a DNZ metamaterial channel can result in significant and uniform amplification that may be employed to enhance material or structural nonlinearities in the channel for applications like transmission switches. This work presents recent analytical and finite element modeling of the use of DNZ channels to enhance acoustic nonlinearities. It also explores and discusses metamaterial mechanisms for attaining a tailored and enhanced nonlinear response.