“So, if you refuse to answer, they can pull you out of the line and over into ‘secondary inspection’ and they can probably hold you there for about 20 minutes or so,” she said. “But they cannot do anything more if you continue to refuse to respond unless something else develops during that time period that would lead to probable cause.”
More than one motorist in the video declined to pull over into secondary inspection, yet they were allowed to go on their way without incident.
.....
So it seems you are within your rights not to answer the Border Patrol agent at an internal checkpoint (this doesn’t go for actual borders!), but the agents are also within their rights to ask you about your citizenship. At least for a while. After that, they’d need probable cause to detain you.

DO SOMETHING!