Ok. I looked into this question *a lot* in the past, although my memory of what I found is a bit old. Here's a try.
You can do ANOVA with random and nested effects in JMP. Once you put the variable into the 'x' box, you highlight it and click on 'attributes' and "random" is an option. But that assumes your y variable has a somewhat reasonable normal distribution.
You can't do other GLMM (poison, binomial) with random effects. It looks like you can when you set up the model within 'fit model'. But, when you hit 'run model', it tells you that the generalized linear model does not take random effects. At least that is the case with JMP 7. But I looked into whether I could do this with a more recent version (JMP 9) and believe the answer is still no.
I turned to Stata and SAS for that analysis (survival with families as a random effect). I found Stata easier to use, but it would be helpful to know SAS or R these days. That was a few years ago, but it seems the same versions of JMP are still available.