She claims these terms aren't racist, but presumably they would be if used of Black people by white people or Asian people? Rather like the "n" word that is bandied around in rap music etc. and presumably in some Black speech. She does admit that the words are offensive… and then immediately defends herself for using them:
'It’s so contradictory it’s unreal – racists have taken these terms and defined them for us. There is no way they are racist. They are meant to make someone feel uncomfortable, but just because something’s offensive doesn’t mean you can’t say it.”'
But you can expect people actually to be offended by them and to question your right to use such language to you with impunity. This is the "I only said it to get a reaction" defence. If you do get a reaction, you have to take the consequences, however.
I'd say (I'm not a lawyer) that this case has nothing to do with whether her ideological beliefs are protected in law. The question concerns the speech she herself used, on a public forum, or that came from a source sanctioned by her (the Race Trust). Is it legitimate for the university to take action against her for speech that she did not issue in her role as advisor? If she is associated with the university qua head/founder of the Race Trust, then that tweet could be grounds for the university to end this relationship, as it would cast doubt on her ability to inspire confidence in others, including students, as an authority on race (reactions ranging from "If this Black woman is using this nasty, degrading term about another Black person, what is she doing telling other people what to do or say about race?" to "Look at the language these Blacks use to one another, why can't we use it too [often said about the 'n' word]?").
The other tweet came from her personally, so may not be relevant, unless she made her connection to the university explicit in the tweet or in her Twitter handle, or perhaps unless her connection is well-known enough for the tweet to bring the university into disrepute.