Kinda 50/50 here. I think the FedGov has constitutional authority to have checkpoints at the border and verify legal citizenship or guest status if they desire. Well inside the border it gets dicey, too much of that starts infringing on people's freedom to go about their business w/o govt hassle.
I agree. This was not at a border, but well within the US. One can refuse to answer questions per the 5th Amendment. Being on the road, if stopped for cause, if he was driving, the only legal thing he has to provide is proof he is legally licensed to drive and his vehicle is legal (registration and insurance). If he is a passenger, he is under no legal obligation to answer any questions or provide any ID.
The checkpoint issue is even more dicey because they are not being stopped for cause. I've gone through many of these ICE checkpoints in Texas and New Mexico and am usually just waived through with out even a question being asked, but if it gets a bit too 'brownshirty' I would be tempted to cite the 5th Amendment.
The person in this case would have best served herself by just asking in response: "Am I being detained, am I free to go?" before getting belligerent as ICE technically can't detain without cause and they wouldn't have cause if you ask this first.