Can independent variables in Poisson model, Negative binomial model and Hurdles model be categorical ?
Yes, they can be categorical ... as in all linear and generalized linear models. In fact, you can continue generalizing/extending the family of regression models, and it will still be true. As long as your model uses a matrix of predictor variables, all the coding trick you have learnt for usual linear regression can be applied. You can represent categorical variables via dummys (also known as one-hot encoding), use polynomials, splines, and so on.
This is the beauty of generalizing regression models, all the tricks you have learnt in the simplest setting can still be applied. You can find many examples in this posts.
Yes it can and you can encode it using an indicator variable. Basically you have a variable that is 1 when the data point is in a certain category and 0 if it is not. Wikipedia provides a nice explanation and example: https://en.wikipedia.org/wiki/Dummy_variable_(statistics)
For a more specific example/elaboration please see here: https://newonlinecourses.science.psu.edu/stat504/node/168/
If your variables are all categorical then the convention is to call this a log-linear model: https://newonlinecourses.science.psu.edu/stat504/node/117/