I have never heard of an employer requiring a doctor's note for only one day absence due to illness. Most employers only require a doctor's note if you are out 3 days. At this point, unfortunately, I think you may have to see a doctor to get documentation of your illness, and the date that you can go back to work.
Call your company and report that you have tried to see your doctor, but can not get in to the doctor until Monday, which means that if they require a doctor's note prior to your return, you will need to be out Sunday and Monday as well. If they complain about your absence those 2 days, make it clear to them that it is their own requirement (doctor's note) that will prevent you from working until Monday after your appointment. How can they expect you to be able to see your doctor on a weekend?
I also suggest that once you do return, you discuss the sick leave policy with supervisors and other employees. It needs to be rewritten. Maybe if you have support of other employees, your company will be more willing to rewrite the policy.
BTW, I support your decision to stay home if you have the flu. I would rather have someone who is that sick stay home, than have them go into work and pass their germs around to others.