We develop a non-equilibrium Green's function formalism to study magnonic spin transport through a strongly anisotropic ferromagnetic insulator contacted by metallic leads. We model the ferromagnetic insulator as a finite-sized one-dimensional spin chain, with metallic contacts at the first and last sites that inject and detect spin in the form of magnons. In the presence of anisotropy, these ferromagnetic magnons become elliptically polarized, and spin conservation is broken. We show that this gives rise to a novel parasitic spin conductance, which becomes dominant at high anisotropy. Moreover, the spin state of the ferromagnet becomes squeezed in the high-anisotropy regime. We show that the squeezing may be globally reduced by the application of a local spin bias.