"University" should be capitalized with Yale because it's the official name: "Yale University" (per the footer in their website).
"University" doesn't need to be capitalized with Harvard, because the official name is "Harvard College" (per the footer in their website). Although they do use the term "Harvard University" throughout their site.
In plural form, "Yale and Harvard universities" is fine. I can't imagine anybody seeing a legitimate fault in that formulation.
Also, if it's already clear from previous sentences or paragraphs that you're talking about higher education, then "universities" may not be needed at all. Consider simply:
"Our expert panelists, including former admissions officers from Yale
and Harvard, will discuss..."