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

I found out that it's a variegated fritillary.

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

Possibly, an Orange striped oak worm - Anisota senatoria