Hyperbolic metamaterials (HMMs) enable control of light beyond what is possible for isotropic ones, as they can support both electric and magnetic resonances in a small volume. The spectrum of a hyperbolic nanosphere has two peaks: a strongly radiative electric dipole resonance and an absorptive magnetic dipole (MD) coupled with an electric quadrupole (EQ). Here, we derive resonance conditions for these modes, elucidating their dependence on the permittivity of the external medium in the quasistatic approximation, including proof of a material resonance of the MD-EQ resonance. The obtained conditions are in good quantitative agreement with numerical simulations and allow the elucidation of the influence of various multilayer parameters on the spectral response of hyperbolic nanoparticles, including their sensitivity to the environment.