I don't know any graduate student who didn't get medical through their university. I only know one postdoc who gets benefits (he is a at a national lab). So I don't know if this is universal or not- but my limited, anecdotal data suggests that graduate students are more likely to get medical benefits than postdocs.
Graduate students at my university were union represented and signed a standard employment contract with the university. Postdocs did not sign such contracts, and were generally at-will employees with few-to-no benefits.