The only pill you should ever consider taking in order to prevent pregnancy after having had sex is the morning-after pill. Are you in a country where it is sold? And, most important, is all of this discussion with him within the time frame for the morning-after pill? It is NOT EFFECTIVE after 72 hours, and it is most effective within 48 hours, of the unprotected sex. If it has been longer than that, there is no pill to take after the fact, that will cause you not to get pregnant. There is no pill that "makes your period come faster." I think your boyfriend is probably pressuring you to take something that he himself does not understand. That is why I asked what pill, and if you have read up on what it is and how it works. Don't take any pill he just walks in and gives you.
I would also give some consideration to getting rid of the boyfriend. He should take more active steps to prevent pregnancy in advance of having sex, such as buying a packet of condoms and putting one on.
This man obviously doesn't want to be a father. Would you really want a man like this as a father to your child. If he is soo concerned about birth control, why doesn't he do something about it. He can get a vasectomy or use a condom , etc. why should you be the one to risk your health?
it doesn't matter if your period comes faster. there still a chance of you getting pregnant if you are not on birth control. and you need to find out what pills he want you to take to make your period start early.
Do you know what he wants you to take? Have you read up on it?