The border patrol can and does regularly put up blockades on major thoroughfares, stops everyone, and potentially could choose to search anyone that they reasonably suspect was recently out of the country or is illegal.
This is not hypothetical. I personally have encountered this when I was stopped and questioned both on the I-5 just north of San Diego, and on secondary streets in Los Angeles. Neither time did they choose to search me, but I saw others that I believe were searched.
As an aside, the ACLU's "Constitution free zone" campaign isn't referring to citizenship checks†; it's referring to a DHS employee's insistence that being within N miles of anything that could be considered a border subjects you to warrantless border searches. Which, again, is something you can find a SCOTUS case refuting.
† Unfortunately, it's now apparently the law of the land that the police can demand identification from citizens, which is one of the great BS decisions of the last 20 years.
I know that people have done so, and the result has been somewhat interesting. Based on what I've read, I suspect that the actual protection from the 5th amendment is currently being interpreted much more narrowly than they were historically, and if the agents pushed it you have a lot less immunity from having to answer their questions than you would like.
Please note that I said, "I suspect". I have read conflicting opinions on this in the past, have no idea which is right, and so it is necessary to put significant disclaimers next to anything that I say on the topic.
On the other hand, the notion that a bunch of uniformed, armed dudes pulling you off the I5 well north of San Diego for no apparent reason and asking things is 'a search you can refuse' is a little glib. When it happens, 'but wait there is case law' or 'I saw this on youtube' is not the first thing that comes to mind.
Reminder, though: we're talking about two different things: warrantless searches and documentation checks. It is, unfortunately, lawful for the police to demand your identification in a wide variety of circumstances, post Hiibel. (You still have the right to remain silent at those checkpoints, which is what the Youtube videos demonstrate).
The reality is that these border patrol checks do happen, often well outside any reasonable range of an actual border and that they tend to be aggressive and searchy and done by people and to people who most likely did not take a close reading of Almeida-Sanchez v. US.
I have been personally stopped near the AZ/Mexico border (maybe 15 miles away), questioned before being allowed to drive away, and I have never actually crossed that particular border.
They key appears to be not to pull to the side of the road for extended questioning, because as long as you're in the main lane, you're holding up traffic.
The Border Patrol is able to operate in that zone, and they operate under somewhat different rules than other forms of law enforcement. Furthermore they sometimes interpret their ability to operate somewhat more generously than a court would agree to. So while in theory they can't do what they sometimes do, in practice they are the ones there with guns and they do it anyways.
(In particular the court has ruled that they can stop and question everyone on a public thoroughfare because the intrusion is short in duration. I've seen documented cases where agents in the field clearly believed that they could question for as long as they wanted to question. The reasoning of the court would not support that conclusion...)
So, very much not really a constitution-free zone. But certainly the border patrol walks closer to the edge of constitutional protections that we take for granted.