Before I make any conclusion, I have to ask you some
questions.
Yous said you have eczema. You mean other and different one then
the rash around your neck?
Have you got this rash also before or is this the first time?
Except the eggs, have you been eating something new what you
haven't been eating before?
Do you wear some new shirt or anything else what can irritate your
skin or use a new laundry agent?
How does the rash look like and except of your neck, where else is
it located and how much? (shape, size, colour, etc)
Usually during an allergy on food, whole body or most parts of the
body will develope a reaction. Anyway, let's see after you answer
those questions above.
Just for the case it's an allergy on some proteine in the eggs,
stop to eat the eggs for one or two weeks and see if it gets
better. I will make a conclusion when I know more about the rash
when you asnwer those questions.
Ofcourse you should visit a doc, especially when you are having
this rash for so much days.