I'm not a constitutional lawyer, but if health concerns can postpone elections, it seems reasonable to me that health concerns can close state borders.
The Texas governor has already been limiting travel from Louisiana, e.g.,
Texas tightens rules on neighboring Louisiana over COVID-19 I haven't heard of any challenges to these policies. At least they haven't made the news, and I think they would.
This issue would be an example of assuming that anything associated with what comes out of Trump's mouth must be all bullshit. Trump's version of the idea is always going to be highly suspect, but there probably are qualified people who have already considered these kinds of issues. I'd like to hear what they have to say.
ETA: I vaguely remember embargoes on California produce because of the Mediterranean fruit fly. That was a long time ago.