My daughter swells up a lot when she gets a bug bite around her eyes. Is this because she is hypersensitive?

Good evening, it's hard to say if your daughter has a specific allergy to mosquitoes when she gets stung near her eyes. What I can tell you is that Benadryl cream is not very helpful in this type of reaction and can even cause reactions when applied to the skin. I recommend that you give Benadryl by mouth if the situation recurs. This will have a better effect on the swelling. Also continue to apply cold water compresses, this is really the right thing to do. It's a pleasure!