If you don't have any eggs you can use Flaxseed or Mayonnaise as an egg substitute for the cornbread.
To use Flaxseed for an egg substitute in cornbread you can use 1 tablespoon whole flaxseeds + 4 tablespoons water (or other liquid) processed together until blended well. 3) 1 teaspoon cornstarch, 3 tablespoons vegetable oil and a 1/4 teaspoon.
When mixed together it makes a good egg substitute for cornbread and other baked goods as well.
You can also just leave out the eggs in the cornbread which I've done before and the cornbread turned out just fine although not as good as it would have if I had used the eggs.
So you can cook and make cornbread without eggs but using eggs makes the cornbread taste much better.