If you're on your soon-to-be ex-spouse's health insurance, it's a good idea to shop for coverage during the divorce process—you'll need to secure your own.
Generally speaking, when health insurance is through an employer, the policy stays with the primary owner, the employee of the company providing it. In some instances, the spouse not employed by the company may have continued workplace coverage for a period of time after divorce.
If you don't have your own work coverage, you may be able to continue your spouse's existing coverage for up to 36 months under the provisions in the Consolidated Omnibus Budget Reconciliation Act (COBRA). COBRA coverage can cost substantially more than employer-sponsored health coverage, as you are entirely responsible for the premiums.
Investigate all of your potential options, including any health insurance that may be offered by your own employer. You may also want to shop your state's health insurance exchange under the Affordable Care Act for a new policy.