One of the reasons teachers take more sick days than you might is really obvious: we get sick more often than you do. I spend my days interacting with 300 children, twenty of those very closely, all school year. As a Kindergarten teacher, my students are young and building immunity to childhood diseases...by catching them.
There is not enough hand sanitizer, hot water and soap in the world to protect me - and if there were, the dry skin from constant washing would lead to new entry points for germs anyway.
Since becoming a teacher, I've contracted classic common colds and flus, immunizations notwithstanding (remember the year they guessed wrong on the virus? I sure do). I've been exposed to conjunctivitis and strep throat more times than I can count - not to mention head lice and fifth disease. I've also managed to get some more exotic illnesses. I missed mononucleosis in college so I could catch it in a Kindergarten (two weeks out of work). I've had whooping cough (the entire winter break with coughing fits to spare for January).
This year, I have missed more days for District-sponsored professional development and events than I have for illness - for whatever reason, this has been a year of Fever and Chills on the Weekend, but Just a Runny Nose by Monday. (Also, I can now teach fairly easily through the annual bout of cold-with-laryngitis.)
Beyond that, I think people forget that teaching as a profession is not sitting at a desk eating muffins. I can't even go to the bathroom when I please. I really need to be in a good, generous mood all the workday long. I spend my day on stage, performing for my loyal audience. And so does every other teacher. This is emotionally wearing work. It leaves you open to illness.
Those Teachers Are So Lazy with The Contracts Bust 'Em Already has been a popular trope lately, making its appearance in the New York Times and the usual teacher-loathing sources. It's offensive.