Generally, no.

When we file bankruptcy, we list all of your creditors (people you owe money to). We even list potential creditors (like that landlord who let you out of your lease early with a sly smile, promising not to sue you). That being said, we don’t list everyone under the sun.
Medical bills are very common in bankruptcy. Whether it’s a chapter 7 or a chapter 13, you want to list the hospital where the medical debts were incurred, as well as each individual doctor, doctor’s office, and ambulance company involved in giving you an outrageous amount of debt.
However, your health insurance provider isn’t one of your creditors. They won’t care about your bankruptcy. They won’t write off your various debts to hospitals in their system because of your bankruptcy. We don’t need to list them.
Now it’s a different story if you are behind on insurance premiums. If you are behind and have an outstanding bill with your health insurance, then yes, we list them. Not for the medical bills and medical debt, but for the insurance premiums you owe them.
Of course, if you’re in doubt, list everyone. It’s better to be safe than sorry.