Scanning probe microscopy and ab initio calculations reveal modifications on the electronic and structural properties of graphene/h-BN heterostructures induced by compression. Using AFM and EFM techniques, with charge injection being made in the heterostructures at different pressures, the charge injection efficiency monotonically decreases with increasing pressure for monolayer-graphene (MLG)+BN heterostructures, indicative of a conductor-insulator electronic transition. Bilayer-graphene (BLG)+BN and trilayer-graphene (TLG)+BN heterostructures show a non-monotonic behavior of charge injection versus pressure, indicative of competing electronic structure modifications. First-principle calculations of these systems indicate a pressure-induced van der Waals-to-covalent interlayer transition, where such interlayer covalent binding, in the presence of water molecules, results in a disordered insulating structure for the MLG + BN case, while it leads to an ordered conducting structure for both BLG + BN and TLG + BN heterostructures. These opposing effects may have a strong influence on graphene/h-BN-based electronic devices and their physics under pressurized environments.