Mechanically driven magnetoelectric antennas are a promising new technology that enable a reduction in antenna size by many orders of magnitude, as compared to conventional antennas. The magnetoelastic coupling in these antennas, a phenomenon playing a direct role in determining performance, has been modeled using approaches that are severely lacking in both accuracy and tractability. In response to this problem, we take a physics-based approach to the analysis of magnetoelastic coupling. We find that certain directions of applied stress will maximize the coupling and we derive general expressions to quantify it. Our results are applied in comprehensive simulations that demonstrate the dynamic nature of the coupling as well as the impact of various operating conditions and material properties. Our work contributes analytical expressions and associated insight that can serve not only as guidelines for the design of mechanically driven magnetoelectric antennas, but also as stepping stones towards the development of more accurate models.