You have not mentioned if your BF is the bio-dad of your four children. If so he should claim them--even if it means the refund goes to child support, since you cannot get a refund, and it would help to pay down his back support. If he is not related to them, then the only way he can claim any of you is if you have all lived with him for the whole year. Then he can get the $500 credit for other dependents for each one of you, which is not a refundable credit, but it does reduce his tax liability. If you lived together all year he can claim you too.
https://ttlc.intuit.com/questions/4499708-what-is-the-500-credit-for-other-dependents-family-tax-cre...
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