How important are eggs for a muffin recipe? I have not actually read Michael Ruhlman's Ratio.  I know that he describes a basic muffin as 2:2:1:1 of flour, liquid, egg and fat.  Prior to discovering Ratio, I only ever thought of a muffin in terms of dry:wet ratios, fat as adding richness, and egg (depending on recipe) as adding richness or structure, so this is new to me.

I am asking because I have not had much success with muffins in the past and am wondering if it is because I have been using recipes without eggs for convenience (because we have usually been out of eggs in this house).

How essential a part do eggs play in a muffin recipe?
There's no simple answer to this other than "it depends". Ratios like the Ruhlman chart are a good jumping off point, but for a lot of chemically leavened things there's a lot more to it. I would highly recommend reading the section on balancing recipes from Cookwise by Shirley Corriher and using it in conjunction with this chart. 

The amount of eggs needed will depend on the type of liquid and the type of flour used. The liquids and flour will also alter how important the eggs are. A good example would be things like banana and pumpkin. In banana or pumpkin muffins, the pureed fruit would be used in place of most of the liquid, but since both of those also contain a lot of pectin, they can also replace the eggs. If the liquid is something acidic like sour cream, yogurt, buttermilk, or cider vinegar, that can also reduce the amount of eggs needed. Some dry ingredients like buckwheat, flax, and chia will also replace eggs. 

In short, the importance of eggs really depends on the specific recipe used. While there may be a good ratio for a plain muffin base, it will definitely need tweaking depending on other ingredients used.