Technically no but I've seen this done a number of ways. If you two are in agreement then that can be done. If you ask the court to decide typically child support and health insurance will be calculated under the guidelines for each parent.
My worry here is that the court may not follow the child's wishes as they may not be in the best interests of the child. Specifically, courts do not like separating siblings. If you are serious about this then you should contact an attorney in the area who is familiar with the judges personalities.
Technically, the amount of support you would each pay would be 20% of your net resources. If there is a disparity in the amount of net resources that each parent has, then it is possible that a court would still order the wealthier parent to pay the difference (or both pay with one check being greater than the other). However, in practical terms, most parents i this situation are able to agree that no one pays child support. Talk with your ex and see if you can work out an agreement that no one pays. More than likely a judge will agree to honor that agreement between the parents.