A mean-field theory is developed for a calculation of the surface free energy of the staggered body-centered solid-on-solid (or six vertex) model as function of the surface orientation and temperature. The model approximately describes surfaces of crystals with nearest neighbor attractions, and next nearest neighbor repulsions. The mean-field free energy is calculated by expressing the model in terms of interacting directed walks on a lattice. The resulting equilibrium shape is very rich with facet boundaries and boundaries between reconstructed and unreconstructed regions, which can be either sharp (first order) or smooth (continuous). In addition, there are tricritical points where a smooth boundary changes into a sharp one, and triple points where three sharp boundaries meet. Finally our numerical results strongly suggest the existence of conical points, at which tangent planes of a finite range of orientations all intersect each other. The thermal evolution of the equilibrium shape in this model shows a strong similarity to that seen experimentally for ionic crystals.