  We study the optical properties of quantum dipole emitters coupled to
hyperbolic metamaterial nano-resonators using a semi-analytical quasinormal
mode approach. We show that coupling to metamaterial nano-resonators can lead
to significant Purcell enhancements that are nearly an order of magnitude
larger than those of plasmonic resonators with comparable geometry. However,
the associated single photon output $\beta$-factors are extremely low (around
10$\%$), far smaller than those of comparable sized metallic resonators
(70$\%$). Using a quasinormal mode expansion of the photon Green function, we
describe how the low $\beta$-factors are due to increased Ohmic quenching
arising from redshifted resonances, larger quality factors, and stronger
confinement of light within the metal. In contrast to current wisdom, these
results suggest that hyperbolic metamaterial nano-structures make poor choices
for single photon sources.
