Hi:
If one checks the Overdispersion test box, the model fit "controls" for overdispersion in binomial and Poisson models by estimating it (the degree of Overdispersion) post-model fit via the Pearson Chi-sq/degrees of freedom.
The JMP documentation is a bit sparse on this topic - see the SAS 9.2 on-line docs for Proc GENMOD for a much better discussion (links in the example below).
If you are so blessed to have access to SAS as well as JMP the example script below runs the JMP Fit Model Platform with GLM personality and provides SAS Proc GENMOD code to match.
Best regards,
-Matt