Yes the Fit Model platform supports the inclusion of both continuous and discrete fixed effect factors. With a mixed model you also have the inclusion of random effects. Fit Model supports mixed models. Add the column to the list of model effects. Select the term and then select the red triangle hotspot labelled attributes. From there you can specify that the effect is random.
Thanks for your response regarding random effects in the Fit Model platform. Random effects are possible when you use the "Standard Least Squares" personality. However, it doesn't seem to work in the "Generalized Linear Model" personality. Please, do you know of a way to use random effects with a Generalized Linear Model in JMP?