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 Youngs 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 $2.5pm0.6$ 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.
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