In the press coverage and on blogs, a lot of people have responded to this story with bewildered remarks like “They don’t really require teachers to shoot people, do they?” (since Kearney-Brown got fired for adding the word “nonviolently”) or, more generally, “What does defending the Constitution have to do with teaching math?” (since most people don’t know that this is a universal requirement for state employees). Some of Cal State’s original statements seemed to be meant to head off that argument, by saying that it’s not about the specific words—it’s just that this is a standard oath and you’re not allowed to change any part of it.
Now, to my non-lawyerly eyes, it looks like the more Cal State and Brown lean on the Smith case, the more they’re making it about the specific words: that is, they’re saying that the word “nonviolently” is incompatible and inconsistent with the duties of a public servant. So it would seem that they do require teachers to shoot people.
Added: Some writers have been referring to this as a “loyalty oath” and wondering whether it goes back to Cold War anticommunism. They’re more or less right. What we have here is an amended version of the 1952 “Levering oath”, which was made mandatory for all public employees after UC got in trouble for firing professors who wouldn’t sign a McCarthy-type loyalty oath. The original version required you to swear that you weren’t “a member of any group advocating overthrow of the government”, but that part was struck down in 1967, leaving just this vague remnant.