I'm not going to do anything on my boyfriend's birthday
Why would you skip his birthday?
Rather than he doesn't like the date, It's because it's not an interesting day for me.
It is kind of rude to not acknowledge your partners birthday or give them a gift. If they are uncomfortable with it you can do a nice gesture on a different day.
It's not my fault that I'm not doing anything on this day. People should respect my choices.
It's one thing if he doesn't enjoy his own birthday, a lot of people have issues with birthdays. But, this sounds like you might be holding him back from celebrating his birthday which isn't being very supportive to someone you love.