  Two-dimensional (2D) magnetic materials with strong magnetostriction are
interesting systems for strain-tuning the magnetization, enabling potential for
realizing spintronic and nanomagnetic devices. Realizing this potential
requires understanding of the magneto-mechanical coupling in the 2D limit. In
this work, we suspend thin Cr$_2$Ge$_2$Te$_6$ layers, creating nanomechanical
membrane resonators. We probe its mechanical and magnetic properties as a
function of temperature and strain. Pronounced signatures of magneto-elastic
coupling are observed in the temperature-dependent resonance frequency of these
membranes near $T_{\rm C}$. We further utilize Cr$_2$Ge$_2$Te$_6$ in
heterostructures with thin layers of WSe$_2$ and FePS$_3$, which have positive
thermal expansion coefficients, to compensate the negative thermal expansion
coefficient of Cr$_2$Ge$_2$Te$_6$ and quantitatively probe the corresponding
$T_{\rm C}$. Finally, we induce a strain of $0.016\%$ in a suspended
heterostructure via electrostatic force and demonstrate a resulting enhancement
of $T_{\rm C}$ by $2.5 \pm 0.6$ K in the absence of an external magnetic field.
