Do we file non resident for CA or MA for a MI resident, paid by a University in California for work done over the summer in a Massachusetts office.

You file non-resident for MA, as the person was physically present there when working. You don't file anything for CA.

MI should allow a credit for taxes paid to MA. Info in this article may help:

https://ttlc.intuit.com/questions/1900021-when-would-i-have-to-file-returns-in-more-than-one-state