bread made with egg safe to leave outside?
Food safety categorizes each food on its own. You cannot take the categories of the ingredients that went in ("flour is shelf stable, eggs need refrigeration...") and decide on the shelf stability of the product based on the ingredients. For detailed information on the basics of how food safety works, read our tag wiki on food safety, https://cooking.stackexchange.com/tags/food-safety/info. 

Breads can be stored on the counter, no matter what they were made with.