  Two-dimensional (2D) magnetic materials with strong magnetostriction, like
Cr$_2$Ge$_2$Te$_6$ (CGT), provide opportunities for tuning their magnetic state
with potential applications in spintronic and magneto-mechanical devices.
However, realizing this potential requires understanding their mechanical
properties, such as the Young's modulus, and the ability to controllably strain
the magnets and monitor their ferromagnetic Curie temperature $T_{\rm C}$ on a
device level. In this work, we suspend thin CGT layers to form nanomechanical
membrane resonators. We then probe the mechanical and magnetic properties of
CGT as a function of temperature and strain by static and dynamic
nanomechanical methods. Pronounced signatures of magneto-elastic coupling are
observed in the temperature-dependent resonance frequency of these membranes at
the $T_{\rm C}$. We further utilize CGT in heterostructures with thin WSe$_2$
and FePS$_3$ layers to control the strain in CGT flakes and quantitatively
probe the transition temperatures of all materials involved. In addition, an
enhancement of $T_{\rm C}$ by $2.5\pm0.6$ K in CGT is realized by electrostatic
force straining the heterostructure of $0.016\%$ in the absence of an external
magnetic field. Nanomechanical strain thus offers a compelling degree of
freedom to probe and control magnetic phase transitions in 2D layered
ferromagnets and heterostructures.
