You sound too immature to get engaged. You broke up with her over a misunderstanding? You dont break up over every little argument.. when you end a relationship you should absolutely mean it!
I dont think its a good idea to get engaged while you and she are long distance. When will the long distance end? When will you be back together properly? I think you should live together for at least 6 months before you propose to be sure you are making the right decision.
I work in a cash for gold shop. You wouldn't believe the amount of women selling engagement rings who never made it to the alter.
"Don't ask a question if you can't handle the answer".