University of Chicago: INxJ
Washington University (St. Louis): ENFJ
I actually thought of typing universities a while back, but thought it may be too difficult since they're made up of students of every type. But those are some of my guesses from a general sense of the student bodies.
University of Florida: ENFP
USC: ESFP or ENFP
MIT: INTP (duh)
Is it that by its indefiniteness it shadows forth the heartless voids and immensities of the universe, and thus stabs us from behind with the thought of annihilation, when beholding the white depths of the milky way?