If you are forced to go to the Doctor for a sick call and you are not already going for a condition (chronic sinus blockage or other recurring issue resulting in sick calls being questioned) it's not a smart idea to just have the company reimburse you for your Co-pay. When you go to the doctor, minor medical center, or hospital the total cost of your visit is not just your $5-20 copay. You are also taking a hit on your yearly payouts and lifetime payouts through your company sponsored health insurance benefits. The company wants a note- fine, but do not be using your insurance for their desires. Your company will reimburse the medical costs so utilize that fact. You are in essence "without insurance" for such visit and will be reimbursed for the entire bill when you call in with the common cold since you are a common human. The company getting slapped with a $500 bill instead of a $20 bill will change their tone, but you are also doing exactly as asked- getting a doctor's note at THEIR expense. You pay for your insurance each paycheck, it is NOT your responsibility to supply the company a note at YOUR cost for their records and full reimbursement.
Think about it.