Is this "Security Info Required" prompt legitimate?

Yes it is. Here's Ars Technica reporting on the matter:

"Apple has begun to ask iOS device owners for the answers to security questions as part of an effort to beef up Apple ID security. iPhone users began widely reporting the change on Thursday, though the changes actually began happening sometime in the last 24 hours. Not all iOS users have seen the request pop up yet, however, leaving some confused about whether the prompt is legitimately from Apple.

The prompt that shows up on an iOS device says "Security Info Required." It proceeds to ask the user to select from a handful of security questions and provide answers."

The screenshot in the article is identical to yours: