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No, public universities are part of the state. They’re not private anything.
The issue was that the Florida law extended to universities, which receive public funding but are privately incorporated. Besides, even if it isn’t a direct “First Amendment” issue, prohibiting professors from saying anything that might make someone feel discomfort because of their race is a HUGE abridgement of speech and academic inquiry.
Besides which, it’s definitely overreaching for the state to force all of its employees to pledge not to support a boycott of Israel in their private life.
Besides which, the Democrats aren’t doing anything to abridge the First Amendment, as far as I can see.