I am interested in the law surrounding jurors. During juror selection (voir dire) a potential jurors can be asked questions to determine their suitability to serve on the jury. Is it ever legal for a juror to refuse to answer any question? Could one, for example, "plead the fifth"?
Yes, there are cases where refusing to respond to the question would be legal. The juror could plead the fifth – so long as he hasn't spilled the beans about what he is trying to protect – which provides for the protection from compelled self-incrimination (any incriminating statement that could be used against you in a criminal charge – civil liability doesn't count here). Anything that is said in voir dire is on the record and under oath. You are effectively witnessing against yourself. As stated on NOLO:
Witnesses can assert the privilege against self-incrimination in civil proceedings as well as criminal ones, despite the seemingly limiting language of the Fifth Amendment. They can assert it in state or federal court, in a wide variety of proceedings (including trials, depositions, administrative law proceedings, and investigatory proceedings like grand jury hearings).
. . .
If, by answering, the witness could provide evidence that might aid the government in prosecuting him, then he has the right to refuse.
There has to be evidence, though, that testimony would subject you to criminal charges. "What is your hair color?" obviously cannot subject you to criminal charges. "Have you ever driven while intoxicated?" Would only be incriminating while the Statute of Limitations has not passed. After that point, you have not 5th amendment protections for having driven under the influence because it will no longer subject you to criminal charges.
Additionally, there are cases in which you could refuse to answer but the court could still compel you to answer. For instance, sometimes questions in voir dire get very personal. If jurors believe a question is too personal, they can try to refuse to answer on those grounds, let the judge know, and the judge would make the decision. If the judge decides they must answer, and they continued to refuse, the judge could hold them in contempt.
On a slightly more practical note, if you are objecting to questions, it will impact whether the attorneys on either side will allow you to stay on as a juror. After an objection to a specific question, the attorney may just decide to nix you.