As well as looking after your own health, please remember that the patients are vulnerable. If due to a carer being unwell and/or tired out and drained and making a mistake as a consequence of that, it could potentially result in injury to a patient and the carer getting into trouble, perhaps even ending up in court.
You mention falling. What happens if you were to get that problem during helping a wobbly elderly patient to move from bed to toilet or wherever.? You could both end up on the floor.
It might be best to log being ill with your doctor, though that might be difficult as it is Saturday.
In reply to your original question. If someone is unwell, it is immaterial how long they have worked there. So you would not be being unreasonable. It is possible that the employer might tell you that you are being unreasonable, but that does not mean that it is unreasonable.
You are a human being. Sometimes human beings get ill. If they did not, our culture would not have doctors, nurses, hospital, sick notes, pharmacies, pandemics, and so on. Such concepts just would not exist.