Neither you nor your chickens can get sick from being cold or wet. You both could get hypothermia if cold enough for long enough but temperature can not cause illness like a cold or the flu or a chicken disease.
You get sick from exposure to viruses or bacteria, some of which happen to be...
How could being in the rain make them sick?
As long as they have somewhere sheltered and dry to go as an option they will be fine. Mine usually choose to just stand in the rain.