We study the transformation properties under metaplectic transformations of generalized Gaussians, which are also known as squeezed coherent states in quantum mechanics. We show that the metaplectic group is acting transitively on these Gaussians, and give a number of explicit formulas. We apply these results to the characterization of Gaussian frames. We state a conjecture relating the Gabor frame property to the symplectic capacity of a certain ellipsoid associated to the lattice.