Are afterburners required to be engaged during catapult launch?
The short answer is very much "no" -- afterburners are not at all required to launch from an aircraft carrier.  Longer answer: Fred Larson's comment is correct: it really depends.  Ultimately a given aircraft, at a given weight, with a given wind-over-the-deck, and a few other things, requires a given amount of thrust to get airborne -- whether it comes from dry thrust or afterburner doesn't change the requirement. 

There are a large number of carrier-based aircraft (the A-7, A-6, and E-2 leap immediately to mind) and there are certainly others that flew off-deck without afterburners.  

Current aircraft like the F/A-18 Hornet can be launched without afterburner in some, but not all, configurations.  For the most part the incremental safety margin and increased take-off weight means that afterburners are used for takeoff, but it's not strictly a requirement.