You cannot apply sunscreen around your eyes. It is true that products used on the face will go through some tests to make sure they are safe. However, these tests cannot secure the safety of products when they are applied around the eyes. In addition, the skin around your eyes is thinner than other parts of the face, so the sunscreen may cause irritation. If you put sunscreen around your eyes and feel uncomfortable, you can rinse your eyes with water. If you really want to protect your eyes from sunlight, you can use a moisturizer or foundation with SPF protection. To protect the skin around your eyes, you should always remember sunglasses which are the most effective protection.
Well, in my opinion, putting some sunscreen under the eyes can be good for sunken eyes. Generally speaking, when someone have sunken eyes, it will just make them look very tired and lose the fresh feeling. And that can be terrible, for women. To treat it, you of course can have some sunscreen which has rich of nutrients good for eye skin. Also, some moisture can be used too. By the way, having more fruits which is full of vitamin c will be effective to sunken eyes too. So just have a try.
I do not suggest you to use sunscreen to around your eyes because of the stimulation on the light eyes skin from the strong anti-sun block character of the sunscreen. In order to prevent you from the sunken eyes, you could wear the sunglasses with good quality. That will work and be effective.