I've always considered eggs as a dairy product, so my answer to the question would be "yes". However, viewing them as a meat product does make sense, though as far as allergies, I'd still consider them to be a dairy product.
If you're looking for a dairy free mayo, there's always Veganaise, or ... this other brand of dairy free mayo that I can't think of the name of.
Kathy... Momma to Rickey (2-6-99), Callie (1-25-01), Isabelle (9-1-06) and Sebastian! (3-4-08)