Health insurance companies won’t allow you to stay on your ex spouse’s health insurance policy once your divorced is finalized. Your options will be to purchase a private policy, obtain coverage for through your employer, or elect to exercise COBRA coverage on you spouse’s plan.
COBRA is typically limited in time and can be quite expensive. In some situations, your spouse can be responsible for payment of COBRA by agreement or court order. In any case, your spouse cannot prevent you from obtaining COBRA through their employer because it is your individual right to elect it.