Well, yes, of course you can apply some sunscreen under your eyes, and it will not hurt your eyes, so you do needn't worry about it. As we know that sunscreen are made of many ingredients, which can be good for the skin. For example, it can just make your skin sticky. But on the other hand, it will also do a help to the eye skin. For example, sunscreen can help to moist the skin under eyes.
You'd better use sunscreen under your eyes which will be bad for your skin of the eyes because of the stimulation on the eyes skins. The eyes part is so weak in the face which you should take great notice when you adopt something. The sunscreen may contain a lot of irritating materials which will stimulate your eyes skin. You could avoid the eyes part. Or else, it may get redness around the eyes.
All right, looks like you are quite afraid of getting your skin tone dark because of sunshine exposure. Anyway, it would be ok to put a bit of suncreen under your eyes, but please try to take care of that. Basically, the sunscreen is used to protect your skin from UV, which makes your skin darker. It simply does no harm to your eyes. So you are good to go. But remember not to get too much tan.