It never would have occurred to me to question egg as an allergen in wine. I stumbled across this topic in the Allergic Living forum.
I decided to contact a winery in Chile, Cono Sur, to get some information from the source.
The winery promptly responded via email and informed me that they do not use any eggs in any part of the winemaking process.
Big sigh of relief! For now, I will continue using Cono Sur wine in my cooking. At the same time, I will be double checking with more of our favorite Chilean wineries.