I went out with my boyfriend for 3 years he broke up with me twice because he thought I was flirting with other guys. In the end I took him back both times, then he broke up with me again for the same reason.I told him that I wouldn't go back out with him even thoug I still liked him... did I do the right thing?

Yes you did, stick to your ground and don't get back with him, he broke up with you for 3 stupid reasons, one you don't even know, sounds like he likes abit of drama in his life and wants you to chase him, he's clearly a jealous person too

https://www.askmehelpdesk.com/teens/if-guy-likes-u-528251.html

https://www.askmehelpdesk.com/relationships/dont-have-good-relationship-boyfriend-someone-else-keeps-askin-me-out-527999.html

End this relationship, and dwell no more on him.

YES girl, your saw your ground and you took it. Most girls won't do that. :)