Red Eye Reduction
The best way to cure red-eye photos is to not use the flash. Barring that, you want a camera that either has a red-eye reduction pre-flash, or a red-eye remover function. Unfortunately, there is no list or search engine that I am aware of that would allow you to search for those camera features.
If you do not mind buying a factory refurbished camera, that would be my recommendation, because you can get a much higher quality camera for a very good price. The following cameras are not as compact as what you may be accustomed to...they won't fit in a shirt pocket...but they have much bigger imaging sensors, meaning you can shoot at higher sensitivities (ISO) without flash and still get sharp looking photos. The Nikon is especially easy to use: