My boyfriend thinks he came inside me and I read that can contain semen. But the next morning I started my period. Am I in the clear?

Two questions, first if he "came" inside you, then it was more than precum, it was all the fluids and would have higher concentration of sperm. But yes in many cases precum or the early fluids would have sperm in it also.

The issue here is that if you have sex the day before you period, you may still have that period and be pregnant, it will not be till the next period time and see if you miss it, that you will know for sure.