A box of white cake mix needs three egg whites. If substituting whole eggs for egg whites, how many whole eggs would I use?
If you truly desire a white cake, you don't want to make this substitution.
The reason the white cake mix directs you to add egg whites only is to to avoid the yellow color from the yolks, which will tint your cake yellowish.
Cake mixes are highly engineered wonders, and are very tolerant of almost everything a home cook can do to go wrong within reason. Still, substituting in the yolks increases the amount of fat—and more importantly, the emulsifier lecithin, and so will change the texture of the outcome. This will probably not be bad, but still will not be the optimal result from your mix.
Ideally you have the weight of egg whites that you require; however I found that a normal sized egg has whites weighing approximately 1.2 ounces.
A whole egg is usually around 2 ounces, erring on the side of less.
Therefore, I would say you may get away with using 2 whole eggs. To be more meticulous, I recommend scaling out 3.6 ounces of whole eggs. You can beat the eggs together and then scale them.
There's also the concern that you won't get the same results by using whole eggs instead of pure egg whites; but considering that boxed cake mix is so robust, I would consider these effects negligible.
The Pillsbury White Cake Mix calls for either 3 whole eggs OR 4 egg whites. Depending on which way you go, the amount of water called for is different - for 4 egg whites you use 1 cup of water, and for 3 whole eggs you use 1 1/4 cups of water. The 1/3 cup of oil is the same for both.
What makes a yellow cake different from a white cake is that you use whole eggs, which gives you a yellow color (from the yolks). If you want a truly white cake, you should use all egg whites, i.e. for a wedding cake.
When I have a lot of wedding cakes to bake, I get the frozen egg whites from my local Bakery Supply. I have a chart on my cupboard which tells me how much egg white to use to equal 1 whole egg, 2 whole eggs, 3 whole eggs, etc.
On the rare occasion that I get an order for a yellow cake, I make a white cake with whole eggs and add a little yellow food color. Clients can't tell the difference, and I'm not going to tell. I find the yellow cake mixes very runny, and don't like to use them.
By the way, Pillsbury Cake Mixes are rated the best by the Culinary Institute of America, as well as Wilton, and they are usually the cheapest.