I am confident that in a just world, the answer should be “no”. But we don’t always live in a just world. I wish I had time to write a careful analysis, but I don’t, so here are some first thoughts. Corrections and amplifications welcomed.
UPDATE: This is why I should never blog from memory, at least pre-coffee. This morning I conflated the right to move about within a state — not a clearly established federal right, cf. the aftermath of Katrina — with the right to move about between states, which is quite well established: Sáenz v. Roe, 526 U.S. 489 (1999) held that a classification (here, lesser welfare benefits for new residents in their first year) that has the effect of imposing penalty on the right to travel violates the Equal Protection Clause absent a compelling governmental interest. What’s more the Court defined the right to travel interstate as having three parts: the right to enter and leave another State; the right to be treated as a welcome visitor while temporarily present in another State; and, for those travelers who elect to become permanent residents, the right to be treated like other citizens of that State.
That ought to settle it.
Three areas of law seem relevant to the question.
First, immigration law, which is a matter of only federal concern. States do not in any way get to control movement across the international border. So if the feds want to let refugees into the US, they are admitted to the US. Even at airports in states that say they don’t want refugees.
Second, anti-discrimination law. Here, we find more of a patchwork. Rather than a single federal statute prohibiting national origin discrimination, we have a collection of piecemeal legislation. It might be that none of the usual anti-discrimination laws, which are aimed at things like housing and employment, speak to the issue of free movement between states.
Third, and not least, there is the Equal Protection clause of the Constitution, backed up by § 1983 of the federal code (§ 1983 prohibits the deprivation of constitutional and federal statutory rights by persons acting under “color of law”). As I understand it, neither the Equal Protection Clause nor any other part of the Constitution has been definitively held by the courts to create a right of free movement between states. [Note the update above – should be “within” not “between”.] Thus, at least at the Supreme Court level, it is technically an open question whether a state might close its borders to everyone. (The question arose after authorities closed a bridge preventing escape from areas of New Orleans hit by Hurricane Katrina. I have some doubts that a state could seal its border legally other than briefly in response to an emergency, but again, if memory serves the Supreme Court hasn’t prohibited that in so many words.) What is not at doubt, however, is that if a state were to attempt some sort of border control, it must do so in a manner that does not discriminate invidiously. (If they are searching for a White Toyota, it is not invidious to only stop White Toyotas.) The state may not treat its own inhabitants better than those of other states. And it may not discriminate on grounds of race or any other suspect class. I would presume national origin is such a class; whether immigration status qualifies (if only a proxy for national origin) is a little trickier, but I’d hope so.
Somewhat related: Important Freedom to Travel Decision From the DC Circuit (7/11/09); Plenty of Blame to Go Round (IV) (9/11/05)
The government’s attempt to get Apple to build a bespoke operating system so they can brute force access to an iPhone without it erasing its data has led the media to some of us who were in the first round of the crypto wars. Today was my turn. A few seconds on 
