We report the pressure-induced structural and magnetic changes in [CuCl(pyz)2](BF4) (pyz = pyrazine) and [CuBr(pyz)2](BF4), two members of a family of three-dimensional coordination polymers based on square mesh {[Cu(pyz)2]2+}n layers. High-pressure X-ray diffraction and density functional theory calculations have been used to investigate the structure-magnetic property relationship. Although structurally robust and almost undeformed within a large pressure range, the {[Cu(pyz)2]2+}n network can be electronically modified by adjusting the interaction of the apical linkers interconnecting the layers, which has strong implications for the magnetic properties. It is then demonstrated that the degree of covalent character of the apical interaction explains the difference in magnetic exchange between the two species. We have also investigated the mechanical deformation of the network induced by nonhydrostatic compression that affects the structure depending on the crystal orientation. The obtained results suggest the existence of "Jahn-Teller frustration" triggered at the highest hydrostatic pressure limit.