Why do my survivors keep getting sick?

I recently found you can change their meal. Click the cook house and their change their menu

At a guess, you could be overworking them or it is too cold in the camp. Check your furnace to make sure it is on and boosted.

Then head to the Chiefs House and start the festivities Edict, that will help the mood!

Upgrade your furnace until they quit getting sick. And make sure they get plenty to eat and upgrade their barracks with heaters.