Nothing to stop you staring in a dogs eyes. If your question is why shouldn't you stare in a dogs eyes, it could be because some dogs would see this as a threat and you should avoid eye contact with an aggressive dog.
On the other hand, by staring at some dogs you can make them back down, although would you want to be the one to decide whether it would work or not?
Also with a nervous dog, it is better to avoid eye contact and offer them a protected part of your body to sniff first, such as a sleeved arm, or bend down and let them sniff your side or back, without trying to touch them.