Erika Mantz, speaking for the University of New Hampshire, decreed, "We believe strongly in the right to free speech as recognized by the First Amendment, and we believe equally in the right of every member of our community to feel safe and respected." By putting those claims side by side and saying they're equally valid, she appears to say that "feeling respected" is a legally enforceable right.
It's logically impossible to enforce a right to feel respected for everyone. Enforcing a right requires preventing or penalizing actions which violate it. If A expresses disrespect for B, then defending B's "right" requires taking some kind of action against A. But this can reasonably make A feel disrespected. There's no way to uphold the "right" of both people to feel respected.
The right of free speech and the right to feel respected can't co-exist. If people can't speak against people who do things they don't respect, they don't have free speech. If UNH officials have a "right to feel respected," the university can and should prohibit and punish any demonstration against its policies or actions. When there's a "right to respect," it's always the people in charge who get first claim on it.
As a government institution, the University of New Hampshire is required to abide by the First Amendment. There is no Constitutional guarantee of a right to feel anything.
It's common for people to talk sloppily about rights to feelings, but UNH has gone further than most, claiming them as having equal status to Constitutional rights.
The right to "feel" safe follows the same analysis, strictly speaking. However, it's a more complicated mix, since it's easy to confuse with the legitimate right not to be endangered by people's actions, so I've left it aside.