For most public universities, I'm pretty sure it would be against the law to willingly take down services. Of course, this doesn't mean students and professors couldn't participate by taking down their sites or protesting.
I'd much rather use Google than the "medical" alternatives that were available a hundred years ago or are still widely used in developing countries. These can result in increased pain, suffering and deaths because of poor diagnosis or idiotic ineffectual traditional cures.
I'd pick Google over a witch doctor, religious medicine or herbal doctor every time!!