I've been a business flyer for nearly 30 years. I've never heard of a sick day after a routine flight, but I did hear the pax behind me recently talking about what I thought was a great, enlightened company implemented by their employer.
Employees get a minimum of 12 hours from the time their flight lands before they're expected to be in their office. Regardless of when the scheduled arrival time was supposed to be. Great for the day after an evening of delays. All that's required is that a message be left informing the office of the situation.