My boyfriend and I have 2 sons. One is non biological and the other is his. I have no income, and he pays for everything off his checks. Can he claim the non biological child as his son? Thanks for any help.

I do believe he can claim him as a dependent/qualifying child whether he is biologically his as long as no one else is claiming him, and I'm pretty sure he would also qualify for the EIC (Earned Income Credit) for him as well.
Here's the number for the IRS--they're really helpful and can answer your questions for certain.
Telephone Assistance for Individuals:
Toll-Free, 1-800-829-1040
Hours of Operation: Monday – Friday, 7:00 a.m. – 10:00 p.m. your local time. Hope that helps. Good luck. :)

Even if your b/f is eligible to claim your child, this does not make him head of household and he does not get EIC as the child is not his qualifying child.

Your b/f can claim your child dependent (as Qualifying Relative) only if:
Your child lived with him/her all the year, and he provided more than half of the child's support.

I believe the BF can file as HoH, as his own son is a "qualifying person." You only need one "qualifying person" to file as HoH. Then he can claim the second child - and you as well - as his dependents under the "qualifying relative" rules - under the conditons that Mukata stated.

Yes, b/f's own son makes him head of household.