Yes, you should use your health insurance to pay for injuries you got in the motor vehicle collision.
Many times hospitals and doctor's offices are reluctant to take your insurance card and make a claim. One reason for that is that insurance companies never pay the full amount of the bill. The hospital and your doctor's office would rather collect the full amount from you rather than a partial amount from your health insurance.
Nevertheless, you should always insist that the hospital and doctor take your insurance card and make a claim on your own health insurance policy.
You or your employer pays for your health insurance and there's no reason why you should not use it. The fact that your need for medical care arose out of a motor vehicle collision which may have been the fault of the other driver is of no consequence whatsoever. You should always claim all benefits available under your health insurance policy.
More than likely, you will also be collecting from other sources such as the medical payment provision of your own insurance policy and the liability insurance carrier which insured the negligent party which caused your injuries.
The fact that you can collect these bills from other sources has nothing to do with your right to collect from your own insurance company.
You should always make claims under your health insurance policy and be sure your insurance company pays all that you're entitled to under the terms of your policy.