Based on the tax code only it is not relevant who pays for the child's health insurance when claiming the child as a dependent on a tax return. However, since the child is covered by state provided health insurance there may be a problem with continuing that insurance if the child is not being claimed as a dependent on your tax return. You will need to get further clarification from the state concerning the child's insurance and the consequences if he is not being claimed by you as your dependent.
This situation is another where being married resolves these types of problems.
