You can, but it may be a lot crumblier than cornbread normally is. Eggs aid in holding baked goods together; in a way, they're a sort of culinary glue. If you have cheese, grate about 3/4 cup and add it to the mix--it will help the cornbread's texture, and if it's a nice, sharp cheddar, it will also make it taste wonderful--especially if you add some finely diced jalapeo, too. (diced onion makes it better yet!)
Many people in the south, especially most of the older generation never add eggs (or sugar either) to cornbread. They will tell you they are making cornbread, not cake. However, the few that do add eggs, will add mayonnaise if they have no eggs.