Embedded eigenstates are nonradiative modes of an open structure with momentum compatible with radiation, yet characterized by unboundedly large Q-factors. Traditionally, these states originate from total destructive interference of radiation from two or more non-orthogonal modes in periodic structures. In this work, we demonstrate a novel class of embedded eigenstates based on Berreman modes in epsilon-near-zero (ENZ) layered materials and propose realistic silicon carbide (SiC) structures supporting high-Q (~10^3) resonances based on these principles. The proposed structures demonstrate strong absorption in a narrow spectral and angular range, giving rise to quasi-coherent and highly directive thermal emission.