We demonstrate coupling to and control over the broadening and dispersion of a mid-infrared leaky mode, known as the Berreman mode, in samples with different dielectric environments. We fabricate subwavelength films of AlN, a mid-infrared epsilon-near-zero material that supports the Berreman mode, on materials with a weakly negative permittivity, strongly negative permittivity, and positive permittivity. Additionally, we incorporate ultra-thin AlN layers into a GaN/AlN heterostructure, engineering the dielectric environment above and below the AlN. In each of the samples, coupling to the Berreman mode is observed in angle-dependent reflection measurements at wavelengths near the longitudinal optical phonon energy. The measured dispersion of the Berreman mode agrees well with numerical modes. Differences in the dispersion and broadening for the different materials is quantified, including a 13 cm-1 red-shift in the energy of the Berreman mode for the heterostructure sample.