Even if your doctor's office isn't open, there should be some sort of after hours nurse line they direct you to. I would call the after hours nurse and explain the symptoms.
It might not be related to anything at Wendy's, it may just be a coincidence.
Also, you say you have cats. Could it be a flea bite? Do your cats have fleas at all?
I'm assuming the kids medication you gave is Benadryl or other brand of anti-histamine. This would be the type of medication you would want to give to help with any kind of allergic reaction or itching.
Edit: Wait, after reading the other response. Is this one single big white spot that she currently has that is big enough to be on her face and arm at the same time? If it's that then take her to the ER or urgent care ASAP. That's not a good thing.