Freedom of speech applies only to the government making speech illegal, and it applies to the states as well, since freedom of speech is a constitutional issue. If a state tried to put penalties on the N word, for example, they would be in violation of the constitution, since the constitution applies to the states (if it didn’t, then old Roy Moore in Alabama wouldn’t be in such hot water:).
Fighting words as far as I understand it were not regulating speech, what it said was that distasteful speech could be used as a defense if, for example, someone called someone’s mother a lowlife, and the other person punched them in the nose, it didn’t make the speech illegal, it said that distasteful speech could be used as a defense for an activity that was criminal, like assault…and it isn’t really a first amendment issue, what it is is the state regulating the consequences of speech.
The other place the government can regulate speech is if the outcome of the speech is found to be detrimental to society (and yes, it can be abused…). Feminists tried to get pornography outlawed, though porn is generally covered by the first amendment, making the argument it ‘harmed women’, that it caused rape, etc, and places in this country tried to pass laws on that based on that argument. It failed in the Supreme Court,not only because it is very hard to define porn (those pressing for the laws put, for example, Playboy magazine on the same level as hard core XXX porn magazines, and also could be used against erotica, the religious right types who were supporting this with the feminists would have made a book like 50 Shades of Gray illegal) but also because the court could find no pressing societal need to do so, the evidence the supporters put in front of the court was mostly supposition and studies of dubious value, and they argued that as distasteful as porn might be, and yes, a negative influence it might be, the burden on restricting a right is very, very high. On the other hand, yelling 'Fire" in a movie theater, or getting in front of a crowd and inciting them via hateful speech to go after a person or a group of people, especially if that group of people then acts on it, will get you put in jail and convicted if they can show that the speech led to the incident.
As far as the private vs public employers free speech gets tricky. A private employer can fire an employee for speech, political or otherwise, that they don’t like. With a government body, like a university, that doesn’t apply, especially to political speech. Because the employer is the government, a new layer is added, because firing or other sanctions on speech would then be covered by the first amendment , and courts have ruled that public employees have rights that private employees do not. In a public setting, they can restrict speech, but it has to be on the basis of showing that there is a broader reason to do so; for example, an employee who openly uses racial epithets or said derogatory things about gay students could be reprimanded or fired (despite what Fox news and the right wing would say about free speech), because speech that interferes with the effectiveness of their job position is not protected, nor should it be, the same way Roy Moore can’t put the 10 commandments or a cross in his courtroom, his first amendment rights are outweighed by his role as a judge in being a fair arbiter, not telling people in front of his court that he is judging based on his faith (doesn’t matter whether he actually does or not, judges and other public officials are supposed to work for all the people, and are supposed to maintain a neutral image to those they serve).
I don’t know if the program will actually work or not, I have no problem with a public service campaign to promote a respectful campus, I would just wish that they ran the campaign by a diverse group of people instead of as likely happened, a couple of people dreampt this up and in the process IMO went a little over the top with some of what they claim is objectionable, much like my story about black ice and master/slave as an engineering term (I told both of these to a friend of mine who works where I do, who happens to be African American, and he thought both of those were the funniest things he had ever heard, he basically shook with laughter and said something to the effect that the people promoting those should get out in the world and actually see how racism works…).