Had a car accident and was in the hospital for close to a week. Health insurance covered everything except copays and deductible. The car insurance is waiting to see if my health insurance is going to ask for reimbursement before they pay out anything.
Seeing that I pay my health insurance premiums for situations like this, can they ask for reimbusement?

They can, and they will, and they're entitled to it. Otherwise, if BOTH policies are paying out, that's "double dipping", and your health carrier can ask YOU for reimbursement.

Even if the other guy was at fault, the health carrier will likely put a lien against any judgement you get against them - so the attorney gets paid first, the health carrier gets paid second, and you get whatever is left over.

The car insurance company of the person who caused the accident is responsible for all medical bills as well.

I am currently settling a case myself. I was injured in an accident caused by someone else and his insurance is going to pay my health insurance back for the medical expenses.