If the child is not your boyfriend's child, then your boyfriend can't claim the child as a dependent unless the child lived in the same home as the boyfriend for all 365 days of the year, AND you don't file a tax return. Even if your boyfriend did claim the child, that type of dependent does not qualify for EIC or the child tax credit.
If it is also your boyfriend's biological child, then either one of you can claim the child but not both.
