  We show how to create quantum squeezed states of magnons and phonons in a
cavity magnomechanical system. The magnons are embodied by a collective motion
of a large number of spins in a macroscopic ferrimagnet, and couple to cavity
microwave photons and phonons (vibrational modes of the ferrimagnet) via the
magnetic dipole interaction and magnetostrictive interaction, respectively. The
cavity is driven by a weak squeezed vacuum field generated by a flux-driven
Josephson parametric amplifier, which is essential to get squeezed states of
the magnons and phonons. We show that the magnons can be prepared in a squeezed
state via the cavity-magnon beamsplitter interaction, and by further driving
the magnon mode with a strong red-detuned microwave field, the phonons are
squeezed. We show optimal parameter regimes for obtaining large squeezing of
the magnons and phonons, which are robust against temperature and could be
realized with experimentally reachable parameters.
