No, since your child is not related to your ex-wife's boyfriend (assuming he hasn't adopted him or gotten married which would make him a step-child), his only option would be a
qualifying relative.
The first rule for a qualifying relative is that the
dependent cannot be a qualifying child of another taxpayer. In addition, a qualifying relative that is not related to you has to live with you the entire year to qualify as a dependent. Since you do have income and had joint custody which means the child lived with you for at least half the year, you would be the person who claims the child as a dependent and all related benefits.
In addition, if your ex-wife is claimed as a dependent, your child is
not a qualifying child of her. You cannot have a qualifying dependent of any sort, if you, yourself, are a dependent.
When entering your child in TurboTax, you should indicate that they lived with you for more than 6 months or else they would not be a qualifying child. The IRS expects you to determine who the
custodial parent is based on who had the child for more nights during the year. However, being that you ex-wife does not file a tax return, is not even eligible to claim the child as a dependent and you have joint custody, the child should be your
qualifying child dependent.
Note: in the event the boyfriend files his return before you do and claims the child, you will not able to e-file your return since the dependent social security number would have already been used. You can mail your return and it should be processed. You will both received letters to provide proof of your circumstances. All you would need to provide is support that the child lived with you for more than 6 months of the year (which 50/50 custody would indicate) and that your ex-wife is a dependent and is not eligible to claim the child. You do not need to prove support - a qualifying child cannot provide more than 50% of their own support; however, the person that claims them, does not need to provide more than 50% themselves. The burden of proof for the boyfriend will be much more difficult. First, he would have to prove the the child lived with him for the entire year (which was not the case) and that he personally provided more than 50% of the child's support. The rules for a qualifying relative are more strict and have a higher burden of proof.