Hydrophobic surfaces with large slip lengths have the potential to enhance electro-osmotic flows. Existing theories of electroosmosis in hydrophobic channels postulate immobile surface charges and/or make a number of simplifying assumptions by considering mostly weakly charged surfaces and thin diffuse layers compared to channel dimension. In this paper, we extend prior models by focusing on planar and cylindrical nanochannels. Our theory accounts for a hydrodynamic slip and a mobility of surface charges, and is valid not only on the scale of the nanochannel with thin diffuse layers, but also on the scale of the overlapping diffuse layer. The model is simple enough to allow us to derive analytical approximations for the electro-osmotic velocities even when the surface potential and charge density are quite large. We also present numerical solutions to validate the analysis and illustrate the variation of electro-osmotic velocities in response to changes in the channel size, potential, surface charge and its mobility, hydrodynamic slip length, and salt concentration. Our results are directly relevant for carbon nanotubes, graphene nanochannels, and conventional nanoporous membranes.