In this work we investigate the self-energy of elliptical dislocation loops in anisotropic crystals and determine the functional dependencies on loop circumference, shape, and dislocation core radius. Systematic numerical calculations using the anisotropic point force Green's function method are carried out with the goal of developing an analytical expression for the self-energy associated with these loops. The resulting formula is shown to accurately predict the self-energies for elliptical loops in anisotropic crystals, as well as the self-energies for simple loop configurations in isotropic crystals, for which analytical expressions exist. We apply this expression to predict the critical shell thickness corresponding to defect-free core/shell nanowires (NW) and further for the first time consider the effect of image energy due to the finite size of NW in anisotropic media using the boundary element method. Consequently, self-energy in NWs is corrected by an energy factor. Moreover, we discuss the dependence of the critical shell thickness on growth direction, with 〈1 1 0〉 NW having the largest, 〈1 1 1〉 NW the next largest, and 〈1 1 2〉 NW the finest.