Eggs are NOT Dairy!
Eggs are not dairy products. Dairy products are produced by the mammary glands of animals and clearly eggs are not. Eggs are safe for someone to eat who does indeed have a milk allergy or intolerance, but keep in mind that individuals with a milk allergy or intolerance is oftentimes allergic or intolerant to other foods as well, including eggs. That does not make it causal, rather just an unfortunate coincidence.