Only one of them can be Head of Household, the other has to file Single. Whoever claims the children will list them on their return and claim all of the related credits (child tax, EIC, dependent care, etc.). You can not split up the tax credits between 2 tax returns. If she claims the children, they go on her return, as dependents and she takes all of the credits. Also, if she claims the children, he needs to remove them from his return. Each child should be listed on only 1 tax return.
They can each claim a child, but since they live in the same household, only one can be Head of Household. The EIC non-dependent status does not apply to an unmarried couple that lives together as they are both custodial parents and no Form 8332 is needed. That status is only for children of divorced or separated parents. @skaw19491
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