Black widow and redback systems are compact binaries in which a millisecond pulsar heats and may even ablate its low-mass companion by its intense wind of relativistic particles and radiation. In such systems, an intrabinary shock can form as a site of particle acceleration and associated nonthermal emission. We model the X-ray and gamma-ray synchrotron and inverse Compton spectral components for select spider binaries, including diffusion, convection, and radiative energy losses in an axially symmetric, steady-state approach. Our new multizone code simultaneously yields energy-dependent light curves and orbital-phase-resolved spectra. Using parameter studies and matching the observed X-ray spectra and light curves, as well as Fermi Large Area Telescope spectra where available, with a synchrotron component, we can constrain certain model parameters. For PSR J1723–2837 these are notably the magnetic field and bulk flow speed of plasma moving along the shock tangent, the shock acceleration efficiency, and the multiplicity and spectrum of pairs accelerated by the pulsar. This affords a more robust prediction of the expected high-energy and very high energy gamma-ray flux. We find that nearby pulsars with hot or flaring companions may be promising targets for the future Cerenkov Telescope Array. Moreover, many spiders are likely to be of significant interest to future MeV-band missions such as AMEGO and e-ASTROGAM.