  Quantum squeezing and entanglement of spins can be used to improve the
sensitivity in quantum metrology. Here we propose a scheme to create collective
coupling of an ensemble of spins to mechanical vibrational mode actuated by an
external magnetic field. We find an evolution time where the mechanical motion
decouples from the spins, and the accumulated geometric phase yields a
squeezing of $5.9~\text{dB}$ for $20$ spins. We also show the creation of a
Greenberger-Horne-Zeilinger spin state for $20$ spins with a fidelity of $\sim
0.62$ at cryogenic temperature. The numerical simulations show that the
geometric-phase based scheme is mostly immune to thermal mechanical noise.
