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, TC. 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 TC in a suspended Cr2Ge2Te6-based heterostructure by 2.5 ± 0.6 K by applying a strain of 0.026% via electrostatic force.