Coulomb drag in double layer graphene systems separated by an h-BN interlayer allows probing of the electron-electron interactions in the effective limit of zero layer separation. Although these interactions can be influenced by plasmons, phonons and exchange and correlation effects, these excitations have never been studied altogether, missing the effects of their coupling on the drag physics. Here we study theoretically the effects of these quasiparticles and their coupling, including also the effects of the electronic exchange and correlation, and demonstrate that the drag resistivity can attain a maximum value at room temperature and beyond, where hybridized plasmon-phonon modes contribute significantly. In particular, the hybridization of the plasmons with the hyperbolic phonons of h-BN, confined within the reststrahlen bands, enhance the drag resistivity. This study paves the way for the exploration of novel many-body physics phenomena in systems coupled through emerging 2D hyperbolic materials.