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: