Is it really not a requirement to answer "Are you a citizen" or "Of which citizen are you a country"?
It seems like from the video that the agents need reasonable suspicion that one is not a citizen? What constitutes reasonable suspicion of not being a citizen? An accent? Skin tone?
Do I have the same rights if I'm a foreign national?
Here's more information about the rights of US citizens: http://www.aclu.org/drug-law-reform-immigrants-rights-racial...
(I know the ACLU can be biased in their views on constitutional questions, but think the answer truthful.)
In general, you don't have to answer questions about your immigration status, though, as the link notes, different "rules apply at international borders and airports, and for individuals on certain nonimmigrant visas, including tourists and business travelers."