I'm afraid I really don't know anything about how to care for moth eggs. If you can't find the info you need via researching on the Internet, I would highly recommend that you post your questions (along with a thumbnail link to the egg post) as a topic in the General Discussion forum. More people are likely to respond that way.

As Bob mentioned, this may or may not be Phyllodesma americana, but it's possible that perhaps the information on that guide page would be useful regarding what the larvae eat. It states:

Caterpillars feed on leaves of alder, birch, oak, poplar, willow, snowbrush (Ceanothus velutinus), chinquapin (Chrysolepis chrysophylla), and members of the rose family

Of course, I think the biggest hurdle is definitely going to be getting the eggs to hatch in the first place. Good luck!

If the eggs end up hatching, I'll try those plants. The only problem is that since it's the middle of winter, most of those trees don't have leaves yet. There are some evergreen oak trees in my yard, but their leaves are very tough and waxy compared to the deciduous ones. I have some roses in my yard that have leaves, but I don't know if the caterpillars would eat those.

That looks like a match, thanks! I still have the moth, so if there are any distinguishing features that need to be photographed I should be able to do that. This is the first January record on here for the genus... I wonder why this one emerged so early.

Two species, Phyllodesma american and P. coturnix are sympatric in California and cannot, according to Powell & Opler, be distinguished by superficial characters. (no rsvp, thanks)

I'll bring it to my entomology professor and see if I can get it preserved so I can look at it under a microscope.