Binders for skillet rice patties I made rice cakes for my toddler: rice, quinoa, shredded potato, eggs, bread crumbs, cheese  onions, corn and cilantro.

If I wanted them to hold together better would it be advisable to add water or egg? To avoid an eggy flavor do I use the yolk or whites?

Thanks
You want to use egg whites for binding. 

Science notes: 

About 1/20,000* eggs has salmonella. For infants &amp; toddlers I would suggest cooking thoroughly or pasteurizing the eggs with sous vide if they won't be fully cooked. 

Egg whites comprise water (88%) and protein (11%, mostly albumin). Egg yolks are water (48%), protein (17%), and fat (33%). (There are some other things that make up the full 100% of each.) The yolk makes up about 33% of the egg.  

The chalazae (the "gross" stringy parts of the egg white that holds the yolk away from the shell) is made from the same protein as the rest of the white. It may look weird but it tastes exactly the same. 

*this is the number estimated by the CDC.