When making a good meatloaf eggs are required to make the meatloaf turn out properly.
Eggs in meatloaf help the meatloaf bind together so it doesn't fall apart and also the eggs add flavor, richness, and moisture to the meatloaf.
If you don't add eggs to the meatloaf the meatloaf will fall apart and won't turn out as good.
So you should always add eggs to the meatloaf recipe if you want to make a good meatloaf.
You can use mayonnaise as a substitute for the eggs in meatloaf since mayonnaise contains eggs and will help the meatloaf bind and turn out good.
I've ran out of eggs before and used mayonnaise when making the meatloaf and it worked out just fine.