I have worked with children for years. Day care, day camps and I taught middle school. All the incidents you have described sound reasonable to me. Kids get dirty, plenty of kids with eczema don't bathe daily, your child got sick. But to an outsider, I can see where they would get the red flags.
Missing a great deal of school can sometimes be an indication that there is an issue at home. Bruises to hide, stories to forget, etc. Maybe financial issues which cause problems with transportation, etc. Your child has missed alot. I imagine if push came to shove, you could document everything with doctor notes.
Switching schools and daycares frequently can also indicate that a person is hiding something and leaves a facility before the truth can be discovered. It can also simply mean you were looking for a good match in child care.
Dirty coat and clothes...it happens. But to someone who is already concerned, it can be an indication of bigger issues. If it's a constant problem of breakfast spilling onto clothes, keep the shirts off until right before you walk out the door.
The bathing issue makes sense to me. Our neighbors have three kids with bad eczema. They don't bathe daily and they don't use perfumed soaps. So again, to an outsider, kids who don't bathe regularly indicate problems at home. Either financial...no water, which can lead to neglect. Or just flat out not bathing the child, because the parent doesn't care. I am NOT accusing you of this, just giving you perspective.
So for someone who cares for children, she is seeing what she is interpreting as red flags. While I commend her for letting you know she was concerned for your child, I am appalled that she would just flat out accuse you of neglect. That is a huge word with lots of implications.
As far as whether you should stay or go...Are you happy with the care your child is receiving? Can you keep returning to a classroom where the teacher is suspicious and will watch every move you make? Will your son adjust ok to a new center? If she reports you to CPS, will that change your mind about staying at that cente?. These are all things to consider while you decide. If it were me and I chose to stay, I think I'd have a meeting with the daycare provider and director and talk about the incident and your expectations.