When you’re answering the questionnaire, you’re already sworn-in and under oath, so I would assume you’d legally have to. Not sure what the penalty would be, though, but I’m not really interested in finding out.
I guess they’d also have to prove it’s yours, though. Still, even though I use a pseudo-anonymous name online, I don’t post anything I wouldn’t want my real name next to.
Edit: OTOH, you could probably refuse to answer which would likely get you dismissed. IANAL, though. The last time I was summoned for jury duty, they didn’t ask about social media accounts or anything like that. Just a few questions that would have indicated whether I could be impartial.