I guess depending on how you open them they do indeed have the right to bear arms

Public university labs are generally public as they're state property (in this particular case, UW Madison is a public state University). Further, recording video or pictures of people in public places is broadly legal in the US. There are only "presumption of privacy" restrictions which apply to places such as bathrooms and private property that is not visible from a public location (ex. a sidewalk).

Obv. IANAL and this is not legal advice.

Generally, in the United States, you can, in fact, just record people. Legally speaking, that is, which doesn't make it a polite or cool thing to do. Necessarily.

If you're on someone else's property, they can of course set any number of rules, and trespass those who break those rules. But even there, recording people, if against the rules, is still not a crime. The crime is trespass, if this journalist we're speaking of sticks around after being trespassed off the property.