With Apple Care can I change the color of my iPhone?

I recently got an iPhone 5, and the only color they had left was black and I could either wait 2 months and get the white or get black (and I desperately need a new phone since mine wouldn't even receive text messages anymore) so I decided to get black and my mom got me Apple Care with it. If I were to break my iPhone at any point I know they'd replace it, and if they replaced it could I ask for a white one? I'm not going to purposely break my phone btw to get a white one… I just want to know for future reference haha.

No, probably not. Simply because they would lose a phone when doing so. Your black iPhone 5 would be repaired and placed into a refurbished section and a white version would be given to you. They would essentially lose a white iPhone. If the phone was completely beyond repair, however, things may be a bit different. But I still doubt they could honor the request simply because of the circumstances.