Yes they are correct. When you say the day I was born on it is pretty possible to change the position of on, either before the clause or after, no change in the meaning. As your former sentence includes word day you can use Which as you can count the day, That as it is alternative to which in some time, When because it is like time clause marker and you can use it with year/month/week/day/....
No change in the main meaning of sentence.
When it comes to not using preposition on, I think, if you talk about your day of birth, it is clear and no need to use on. But it is informal, I think