Yes, exercise can make your eyes get whiter because of the moisture role. If you do the exercise a lot, your metabolism process will increase. Your sweat will be expelled from the body easily. Your eyes will become bright. That is why the eyes get whiter. In order to protect your eyes, you could also eat more food with vitamin C which could help you get this effect.
It is true but hard to discover. I always do some exercises and I have the same situation which is normal. When we do sports( I mean the plenty one), you can feel out of oxygen and even dazzling, that's because your heart can not supply the blood you need right now. Without enough blood flowing in the vessels of your eyes, it will seem whiter in the surface. As eye white is shallow in color, it's hard to discover the change. However, which you can obviously see in your lips( red turns to whiter).
Hope that will help.