A couple thoughts for you:
1. You might want to try the Generalized Linear Model personality in the Fit Model platform. You can specify the response distribution (Poisson is an option) and link function (log is one of the options) as part of the model specification window. Here's the link to the JMP online documentation for the Generalized Linear Model personality:
https://www.jmp.com/support/help/14-1/generalized-linear-models.shtml#
2. If you have JMP Pro, within the Fit Model, Generalized Regression personality, there is a zero inflated Poisson distribution option for situations such as you describe...a predominance of zeros within the response population. Here's the link to the JMP Pro online documentation for the Generalized Regression personality distribution notes:
https://www.jmp.com/support/help/14-1/distribution.shtml