I discovered the culprit while digging around in the Java objects of a GUI under development. If you use findjobj to retrieve the Java object corresponding to an HG object, it may cause the menu flashing behavior I was seeing. In particular, it happens when findjobj is processing the menu objects. This can be disabled using the 'nomenu' option.
I was seeing menu flashes when calling findjobj(gcbf, 'depth', 0) to get the Java frame for the current figure. Changing that to findjobj(gcbf, 'nomenu', 'depth', 0) eliminated the flash.