I can't answer, but you might want to post over on the Butterfly Garden forum, if you don't get a response here...

The round, yellow/orange egg on an orange tree sounds like a giant swallowtail, but the description you give of the caterpillar doesn't, so I assume it's some type of moth. Just in case, here's a picture of a giant swallowtail hatchling - Sherry

Somewhere in this house I have a book on US Insect Pests and that caterpillar is on the cover. I can't find the book but I do remember not being impressed with the moth it made. I see them every year. By the time they are this big they are almost done growing and soon will climb down and pupate.

Apatelodes torrefacta I have a little beginner's book about identifying caterpillars that I can't find anywhere! But it shows the most common, most likely to be seen, caterpillars in the eastern U.S., as I recall. And I think it's by Peterson. If I were you, I'd get it, then, when you see a caterpillar that you can't ID, go to that book first. If you can't find your cat, find one in the books that has similar characteristics and look up caterpillars in that family. I guess I'll have to order another one of those helpful little books. The common name of your cat is spotted apatelodes. They come in white and yellow, I've seen both. Sherry