"君に会うと、会いたくなくなる。" means "If I meet you, I will start wishing not to meet you."
In parts 君に会うと. The と is one of your four many options for if type expressions. But this particular one has a 前後 (before/after) connection some of the others lack. This means the event following in this case happens temporally after. So it could also be translated "After I meet with you, I start to wish I didn't meet you" or something roughly like that (the tense in the ～と clause is immaterial in Japanese).
The second half involves the following sequence:
会う to meet
会いたい to want to meet
あいたくない to not want to meet
あいたくなく adverbialization of the constructon
なる in this case to become
ergo, I begin to not want to meet you.
Thus, together, whenever I meet with you, I start wishing I didn't meet you.