Many problems in science and engineering are described in terms of equilibrium shapes on which certain conditions are imposed and which separate regions where the solution may have different properties. A prototypical problem of this type involves inviscid vortex equilibria in 2D and axisymmetric 3D geometries characterized by compact vortex regions embedded in a potential flow. Computation of such equilibrium configurations is made difficult by the fact that it requires finding the shape of the boundary separating the two regions. Similarly, studying the linear stability of such free-boundary problems is also challenging as it requires characterization of the sensitivity of the equilibrium solutions with respect to suitable perturbations of the boundary. We will demonstrate that such questions can be in fact systematically addressed using techniques of "shape calculus" applied to the boundary-integral formulations of such problems, leading to elegant and accurate computational approaches. In the context of vortex dynamics we use these techniques to efficiently compute the family of inviscid vortex rings initially discovered by Norbury (1973). We also obtain an equation characterizing the stability of general vortex equilibria from which certain classical results of vortex stability can be derived as special cases. Finally, this approach is employed to solve open problems concerning the linear stability of Hill's and Norbury's vortices to 3D axisymmetric perturbations, which leads to some unexpected findings.