I don’t know what is “correct”, but Why do you care what I wear is by far the most common way to say it. In my view, if that’s the way everyone says it, it automatically makes it correct.
The sentence Why do you care about what I wear? is more common than it is without ‘about’. However, the absence of the preposition does not affect the meaning intended.
If this is a calm sentence, such as “Why do you care about the French Revolution “, you would include “about””. This sounds like it’s an emotional response to an intrusion on their personal business and “about” just seems to get in the way.
You might also encounter “What do you care what I wear?” Or “Whaddaya care?” All of these are taking umbrage with respect to a previous statement.
BTW, the movie is Something’s Gotta Give ( 2003). I love it!