Two-dimensional magnetic materials with strong magnetostriction are attractive systems for realizing strain-tuning of the magnetization in spintronic and nanomagnetic devices. This requires an understanding of the magneto-mechanical coupling in these materials. In this work, we suspend thin Cr₂Ge₂Te₆ layers and their heterostructures, creating ferromagnetic nanomechanical membrane resonators. We probe their mechanical and magnetic properties as a function of temperature and strain by observing magneto-elastic signatures in the temperature-dependent resonance frequency near the Curie temperature, T_C. We compensate for the negative thermal expansion coefficient of Cr₂Ge₂Te₆ by fabricating heterostructures with thin layers of WSe₂ and antiferromagnetic FePS₃, which have positive thermal expansion coefficients. Thus we demonstrate the possibility of probing multiple magnetic phase transitions in a single heterostructure. Finally, we demonstrate a strain-induced enhancement of T_C in a suspended Cr₂Ge₂Te₆-based heterostructure by 2.5 ± 0.6 K by applying a strain of 0.026% via electrostatic force.