You must have a career services/advising office. You are not the first international premed at your UC. Go ask.
Go to the hospitals associated with UC medical school and ask if you can volunteer there as an undergrad pre-med student. I do not see why a non-resident cannot volunteer. Ask your pre-health advisor. They will help you immediately.
That is not true - I was also an international pre-med at a major US college (East coast, Ivy) and this was my experience:
1.) Although I was not the first international student at my school, my pre-medical advisers had nothing to tell me that would be different from their usual US-resident pre-meds. They do not really collect extra info for international students. Most will only tell you that it's close to impossible to do and that's where their international-specific advising ends (to be fair, I also got a contact for one of the previous international pre-meds, but that was it) - which is why I started my blog (see my signature).
2.) The resource (internal at my school) they referred me to - which is common to all pre-meds, not just internationals - was a list of various internships and clinical experiences, all of which sounded amazing. When I went over those, planning to apply to some of them for the upcoming summer, I found out most of them required US citizens or GC holders - and after I applied to bunch of them, at least half of those responded right away that I was ineligible due to my citizenship/visa status and most of the rest I never heard back from (and I do consider myself a competitive applicant). The same happened when I was applying for positions for my gap year.
If your school has a similar database, you can try to use it, but look carefully and potentially contact the employers about the requirements before applying. Try to ask people you know - other pre-meds (whose parents are often MDs) or MDs themselves, if they would know about anything (that's how I ended up getting my summer position). But be ready to spend a lot of time on it, it is not easy, indeed. Another option is also to look around the extracurricular clubs and hospitals affiliated with your school - both of which will usually take you as an affiliate with the university, not caring about your citizenship. (I was volunteering in nursing homes through a student club.)
If you are only looking for an interesting experience, another option would be trying research lab positions, where they are more used to internationals (which is what I ended up doing for the gap year), but with clinical positions, it is really hard. If you figure out an easy way to do it, please share, I would be happy to know, as many other pre-meds ask me the same question! Good luck!!