As @txnelson commented, there is no test option natively in JMP currently (as of 17.0.0) for a "multiple comparison's version" of a test for Unequal Variances. You can certainly do the variance comparisons separately pairwise, but you won't get the benefit of alpha correction like you do with a multiple comparisons test on the means with a method like Tukey-HSD.
I'd recommend searching the Wishlist and putting this one on there (community.jmp.com/wishlist) if it's not there already.
Note that in JMP 17 in Oneway, you now get a plot of the 2-sided confidence interval of the standard deviation by Level, vs the overall average standard deviation (dashed black line). This is meant to give you a graphical-visual indication of where the variance might be different between levels (relative to the grand average standard deviation).
You can also use Analysis of Means (ANOM) Methods (e.g. ANOM for Variances under the red triangle in Oneway), or Heterogeneity of Variance (HOV) testing (in the Variability/Gauge Chart Platform) to help explore/ identify where specific variances are not the same between groups.