In what? Not, for instance, in scrambled eggs. Not really in poached eggs. And certainly not in green eggs & ham.

We are really hard pressed to come up with a dish in which you could simply substitute mayonnaise for eggs. Yes, mayonnaise has egg yolks in it, but it has a lot more oil than egg, and the texture and binding properties and taste that eggs bring to a dish would not show up if you used mayonnaise. If you'd like to share with us the recipe for which you'd like to make the switch, we'll try to give you a reasoned and well-considered opinion.