Can any regulation of speech that is not viewpoint-neutral be legitimate? How does this link to permissibility of campus hate speech regulations?

There are some legal consequences to certain kinds of
speech under various state and federal anti-discrimination laws, and this has passed
muster under the United States Constitution and state constitutions.  Also, some states
now have enhanced sentencing for hate crimes, which means that it is likely some speech
is involved.  I don't know if any of those has been challenged under the First
Amendment.

For the most part, anti-discrimination statutes
do not have criminal consequences and usually involve what is called a "make whole"
remedy, meaning the law is meant to put the victim in the position he or she would have
been in had the unlawful discrimination not happened.  Using the "n" word to a co-worker
would be one example of a violation of such a statute.  Consequences in a case like this
could very well be losing one's job.

Most campus policies
are pretty much in line with anti-discrimination statutes, so are not likely to be
challenged.  Also bear in mind that the First Amendment is meant to stop the government
from repressing speech.  So a private university is not subject to the First Amendment
at all.  Similarly a non-governmental workplace is not subject to the First Amendment,
and a company is perfectly within its rights to tell its employees what they can and
cannot say.