Can my boyfriend adopt my son even know he has lost his parental rights to his other child and we are not married?

Probably not.

Most states require marriage or some sort of long-term commitment (a marriage LICENSE is good enough some places, so that an adoption can take place at the same time a marriage does---to create a "family").

You will need to see a family attorney in your area to find out for sure what the requirements are for him to adopt. The biggest thing is this: will the biological father relinquish his rights?