Anisotropic magneto-electro-elastic (MEE) materials exhibit strongly coupled mechanical, electric, and magnetic responses, making defect analysis more involved than in purely elastic media. Existing numerical approaches for polygonal holes in MEE materials are predominantly based on the finite element method (FEM), which requires full-domain discretization and intensive mesh refinement to obtain convergent results. To provide the exact solutions or possibly more efficient numerical solutions, in this study we explore the associated analytical solutions and boundary element analysis. Building upon the newly obtained solutions for an arbitrary polygonal hole in anisotropic elastic media [1], we, for the first time, extend them to the corresponding problems of MEE materials by incorporating fully coupled constitutive relations and supporting the related electrical/magnetic boundary conditions. A critical step is the extension of the constitutive model to a fully coupled MEE description, where mechanical stress–strain is consistently linked with electric displacement–electric field and magnetic flux density–magnetic field through a unified constitutive matrix. By this way, two versions of Green’s functions for polygonal holes in MEE media are derived using the complex variable Stroh formalism [2]. One is derived by the perturbation method with conformal mapping, and the other is by nonconformal mapping. Using these two Green’s function as the special fundamental solutions of boundary element method (BEM), a hybrid BEM (HBEM) proposed previously for the polygonal holes in elastic media is now extended to the cases with MEE media. Similarly, in this study two versions of analytical solutions are also obtained for an arbitrary polygonal hole in MEE media under uniform loading at infinity. The analytical solutions and HBEM presented in this paper are then verified through comparison with Ansys (FEM software), since it does not provide direct MEE capability, MEE is validated via reduced cases of piezoelectric media.