I’m not a lawyer in any jurisdiction, but I think your intuition is correct, that the First and Fourth Amendments would come into play in this sort of situation, although the question is how deep are the attorneys allowed to pry before running afoul of these constitutional protections.

The First Amendment deals with free speech without interference from the government, and is interpreted to include anonymous speech, with only small exceptions when it intersects with other enumerated rights. However, the cases dealing with anonymous speech tend to be related to whether to unmask the person behind an anonymous speech, rather than trying to discover a person’s anonymous writings without initially suspecting that they had any. One such example case relates to the Devin Nunes’ Cow Twitter account, where a defamation suit brought by the former House Representative tried to force Twitter to reveal the account’s owner. Ultimately, a judge removed Twitter from the suit, so the owner remained anonymous.

The thrust of protecting anonymous speech is much the same as for whistleblowers: people are less willing to speak the truth if they know the haters will seek reprisals against them. This is a chilling effect on speech, which the First Amendment frowns upon. Likewise, it can be argued that the forced, after-the-fact revealing of an anonymous account will deter current and future anon account owners from posting candidly, even if their posts are perfectly legal.

Separately, the Fourth Amendment deals with rights against search and seizure, allowing only reasonable searches following particular processes. This right combats the problem of results-oriented, overzealous govt officials searching every person until they find a crime, rather than identifying specific persons for whom may have committed a particular crime. If the homes of 100 random people were searched, it’s likely they’d all have something they’d want to keep hidden, either for personal or social reasons, or even criminal in nature. But the Fourth prohibits this sort of fishing expedition.

But the Fourth is not a total bar to searches: they just have to be reasonable, which can vary based on circumstance. For example, if someone matches a crime victim’s sworn description of their attacker, and a police officer detains that person on the street, a pat-down search of that person is reasonable to check for weapons, like a hidden gun or knife, which could harm the officer. It would not be reasonable to also search for evidence of other crimes, such as for drugs or counterfeit money, because probable cause did not exist for those crimes. Likewise, if this pat-down occurred instead after the TSA checkpoint at an airport, then that search for weapons may be unreasonable, because there shouldn’t be any way to bring a weapon into that part of the airport.

A question to a jury candidate arguably meets the reasonable criteria, at least initially, because there’s a bona fide reason why the question is being asked: to assemble an impartial jury. But this can quickly become unreasonable if the attorney questions stray off into the weeds. The reasonable test is very “fact intensive”, meaning every case is different and there are no hard-and-fast rules that can be easily applied.

That said, I think you were thinking of the Fifth Amendment, which among other things deals with self-incrimination. That is, the government cannot force a statement from someone, and then use that statement against that person to punish them. This came about because beating confessions out of people is a Bad Thing, although it’s notable that the Fifth doesn’t actually prevent barbarous behavior. Merely, it means the answers cannot be admissable toward punishing somebody.

The Fifth is strange in that it’s the government that has the choice: either 1) compel someone to talk, but they will suffer zero criminal consequences for what they said, or 2) don’t force them to speak, and thus allowed to pursue any/all lawful penalties. The first situation arises commonly in testimony before non-judicial bodies, like the US Congress. For example, a House committee might weigh the pros and cons of quizzing a former mob boss or a tech company CEO, and decide that their truthful answers – compelled by time in a jail cell if they don’t speak up – is more important than them getting their comeuppance in court later. Often times, the testimony obtained can be used against a different person, which is perfectly admissable because it’s not self-incriminating. Because the govt has the choice of these two options, it’s possible to compel someone to admit to embarrassing details, so long as the threat of legal prosecution is zero, usually guaranteed by a grant of immunity. It’s not clear to me if immunity also carries over to civil lawsuits filed by victims later.

So in a jury selection process, it’s possible for the court to compel the jury candidate to answer about their anonymous social media account, but if a follow-up investigation by a prosecutor finds that the account is somehow criminal, the prosecutor is barred from pursuing charges, by the Fifth Amendment.

In summary, in the scenario where an attorney asks “do you use social media, and describe your accounts”, the question is likely proper per the Fourth Amendment. But an answer such as “I have an anonymous account about cat memes” is likely sufficiently responsive, if it’s the truth. However, the follow-up question of “what is the name of your anonymous account” is likely improper because of the First Amendment. Again, in most courtrooms, the judge would probably take the hint and figure out a way to excuse the jury candidate before it got this invasive.