You can analyze categorical predictors in regression models--including those using Poisson-distributed outcomes. You just need to ensure that you code the categorical variable appropriately (e.g., using dummy, effect, or contrast codes) in order to fit a model that yields interpretable/sensible intercept and slope estimates. Good descriptions of the various coding strategies can be found in West et al. (1996) and Cohen et al. (2013)
References
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2013). Applied multiple regression/correlation analysis for the behavioral sciences. Routledge.
West, S. G., Aiken, L. S., & Krull, J. L. (1996). Experimental personality designs: Analyzing categorical by continuous variable interactions. Journal of personality, 64(1), 1-48.