How important are eggs for 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.