Are afterburners required to be engaged during catapult launch? Is it required to engage afterburners during catapult launches from an aircraft carrier?

This answer mentions that afterburners are used for night takeoffs as per regulations, but not required during day takeoffs. Is this true? If yes, why?
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.