Do you or any members of your immediate family own a gun? If so, provide complete ownership and registration information. Has the registration ever lapsed? Please also describe how and by whom it is used and whether it has been the cause of any personal injuries or property damage.
(See the New York Times article.) Naturally, the Administration is entitled to know a great deal about its prospective high-ranking members, especially with regard to possible violations of the law, both for its own sake and in order to anticipate possible political problems. And the questionnaire certainly asks about a lot more than just guns, e.g., question 46, "Have any members of your family or close social or business associates been arrested for, charged with and/or convicted of a crime, other than a minor traffic violation? If so, please identify and describe each such arrest, charge or conviction. Please provide the same information for anyone under your professional supervision, or anyone of your superiors." Still, it struck me as interesting that gun ownership and gun registration was one of the things that they'd specifically ask about. It also struck me as interesting, though perhaps necessary given the reasonable worry about political blowback should the facts come out, that people would be told to report (1) gun ownership not just by themselves but also by immediate family members, and (2) possible hitherto undisclosed criminal conduct by their immediate family members (which likely includes adult children and possibly parents or siblings, though that's hard to tell).
UPDATE: As some commentators have pointed out, most states don't require registration for guns, but I understood the question as asking about registration information, if any: If the guns don't need to be registered and have never had to be registered (because the people had only owned guns in no-registration jurisdictions), then the registration information would be "no registration required," and the answer to "has the registration ever lapsed?" would be "no."