Yes, a U.S. citizen has Fifth Amendment rights at the border. Statements you make to law enforcement can be suppressed if you were in custody at the time the statements were made, the questioning amounted to interrogation, and you were not given Miranda warnings or requested a lawyer.
However, for routine questioning at the border, federal courts have said that Immigration rules and regulations permit border questioning as a routine and necessary predicate to enforcement of immigration laws. United States v. Silva (2d Cir.1983); United States v. Moody, (2d Cir.1981). In these cases, the court said that the authorities can ask questions related to the admissibility of the individual and that person's belongings into the country and Miranda warnings are not necessary for this questioning.
Furthermore, if you are simply being asked routine questions for entry, you are not considered to be in custody and the 5th Amendment does not apply. The Fifth Amendment right to remain silent applies to questioning only when you are in custody. This is the case at the border as well as elsewhere in the U.S.
If you are being processed administratively and are being asked questions that are intended to elicit an incriminating response, even if you feel like you are not free to leave, the right to remain silent may not apply because you are only in administrative custody to determine if an arrest is warranted for further criminal investigation. U.S. v. Diaz -Quitana.