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.