If I understand your question, no.
Only one person may claim someone else as a dependent. If you are living together unmarried with the other parent, only one parent can claim the child as a dependent and that parent gets all the tax benefits. They should answer "no" to the question about a custody agreement because that only relates to a legal agreement between parents who live apart. The other parent should not even list the child on their return, to prevent problems. The rules for splitting the benefits of a dependent (which creates the label "nondependent" in the program) only apply to parents who are divorced or separated and living apart and do not apply to unmarried parents living together.
So basically, if the parent who pays more than half the expense claims the child, they probably qualify to also file as HOH, and the other parent files single with no dependents. If the parent who pays less than half the expense claims the child, they file single, and the parent who pays more than half also files single because they have no dependent to qualify for HOH.
