At an inland checkpoint, vehicles are stopped without individualized suspicion. The question quotes an unsourced statement from a Wikipedia article that US citizens are not required to produce documentation at such a checkpoint, but this statement is not necessarily correct.
If someone claims to a Border Patrol officer that he is a US citizen, the officer can either accept the claim or not. If the officer has a reasonable and articulable suspicion that the person is not actually a US citizen, the officer can continue to detain the person for the purpose of investigating the claim; this investigation can include demands for identification documents.
Furthermore, even if the officer accepts the claim of US citizenship, the officer is nonetheless empowered to investigate other crimes which he has a reasonable articulable suspicion are being committed or are about to be committed.
Finally, if the subject of an investigation disputes the existence of reasonable suspicion, the way to resolve that dispute is not by arguing with the officer. The only place where such a dispute may be resolved is in a courtroom. One can assert to the officer that there's no reasonable suspicion, but if someone refuses to show ID when the officer demands it, that person risks being arrested for impeding the officer's performance of his duties. At that point, you're just gambling that the officer's claim of reasonable suspicion won't convince the court -- but maybe it will. And whether or not it does, you've gotten yourself arrested.
Do these drivers have a solid legal basis - stating that they have been illegally stopped for no reason, that there can be no probable cause solely by driving near a border, that the constitution forbids these kinds of stops - for refusing to show ID?
The stops are legal, as decided by the Supreme Court in United States v. Martinez Fuerte in 1976. The officers do not need probable cause to stop vehicles at a checkpoint. The constitution does not forbid the stops, because the Supreme Court has ruled that the constitution permits them.
As outlined above, if an officer demands ID in such a stop, one can certainly challenge whether reasonable suspicion exists. It is wise to clarify whether the officer is demanding ID or merely requesting it. If the officer says the production of ID is required, the subject should make it clear that he is cooperating only because of that fact, and is not producing the ID voluntarily. The same approach would apply if the officer wants to search the person or the vehicle.
Can they legally refuse to show documents and refuse to answer questions as to where they may be traveling to and from? They don't appear to be getting arrested.
In general, yes, it's legal to refuse to show documents unless there's reasonable suspicion of a crime. It's legal to refuse to answer questions even if there is a crime, because of the fifth amendment protection against self incrimination. However, as noted above, in the moment of the stop itself, it's impossible to know whether or not the officer really has reasonable suspicion of a crime. For all you know, someone driving a car identical to yours may be wanted for a crime -- that by itself would create reasonable suspicion even if you are in fact totally innocent.
Can only US citizens refuse? What about Green card holders?
A green card holder can claim that status when asked by the officer. Just as with a US citizen, the officer can accept the claim or ask for evidence to support it. With green card holders, however, there's an additional wrinkle, because unlike US citizens, they are in fact required to carry their green cards on their persons by 8 USC 1304(e); it is a misdemeanor not to have it, and failing to provide it could in fact constitute reasonable suspicion.
How does the Border Patrol know determine they may or may not be citizens if the individuals are successful at refusing to show ID?
They can't really know. All they can do is decide whether they have reasonable grounds to suspect that the person isn't a US citizen. The people that you see on YouTube being allowed to proceed without showing ID are people for whom the officers have determined there is no reason to suspect that they are not US citizens.