The "accepted" interpretation is that the 1st amendment right to freedom of speech only applies to government agencies giving punitive sentences for speech. See for instance, China-- where if you you are openly critical of the the party, or of government, you disappear.
That is not permitted in the US. However, there is no compulsion to overlook offensive speech from normal citizens, which HR people happen to be.
EG, if you and I are both citizens, and I call you a "Fucking moron" to your face, you are well within your rights to consider me a bad person, and not want to associate with me, or consider me as a candidate if for whatever reason, I come seeking employment from you, etc. That is a kind of defacto punishment for speech, that is a natural consequence of said speech. This is what the video likely is referring to, and is what is alluded to in the instance of the HR drone reading the hot slathering excrement smeared all over your facebook page, and deciding that "you aren't a good fit." For example, finding a whole lot of racially charged epithets/messages, for a candidate applying to work at a highly culturally diverse workplace.
I would personally prefer NOT to allow HR drones to be permitted to read facebook pages as part of their hiring process, (Since people like me are defacto punished FOR NOT HAVING ONE, because of how heavily they have come to rely on the things for making such value judgments, and when they find "OMG! He does not have one!" they go into paranoid mode, and refuse to hire you, even if you are a perfect gentleman.) but that is like pissing into the wind.
The matter of contention is still that the GOVERNMENT cannot punish you, not that you are free from the consequences of the speech-- which is exactly what was being discussed.