In my opinion, the pressure to change should not be placed solely upon your son. His personality is more sensitive than others and it will make some things more difficult but that doesn't mean he should be the one to change. By telling him "the more you cry, the more people will tease you," you can be reinforcing an attitude that the teasing is his fault - it's not.
I grew up being a more sensitive child and endured teasing of various magnitudes throughout my school years. What helped the most was having parents who understood who I was and defended and guided me through difficult days.
My recommendation is to give your son permission to cry - it's okay for him to express his emotions by crying. When he is teased, help him process what is happening and take action where necessary to minimize teasing.
For example, when visiting one of his cousins, he would say it's time for my son to go home and my son would start crying.
Remind your son that the only person/people who will tell him it's time to go is you. He doesn't have to listen to anyone else. Then address the behavior of his cousins because that is what truly needs to change.