I love eye contact; it is the "connection" that a human and canine shares. My dogs often look for eye contact either for guidance, reassuance or for communication. Luke is my big "eye contact" dog. Just yesterday while we walked the beach with a couple of other dogs; he checked in often. He will look at me until I look at him and we connect; then he is good and off he goes.
Dogs communicate with eye contact; if you could see my Jack Russell when someone is eating. She is not the type to just jump up and grab your food; actually yes she is but she has been trained not to do that. So; she will sit beside you and look deeply into your eyes; then with intensity move her eyes to the food. This is her way of saying "give me that food, that I am looking at." Her eyes move purposefully from the eaters eyes to the food; back and forth. She is communicating her desires.
Tilley speaks volumes with her eyes; she always has. Even when I was in the process of choosing her out of the litter she was using her eyes; more so than any other puppy I had seen up to that point. It was this intense eye contact that made her stand out of the crowd and to this day she still uses this eye contact to communicate to us. Often I get wrapped up in my computer work and forget time. If it is past Tilley's meal time she will come and stare at me; she has done this for so long now that usually I notice right away. If I do not notice immediately she will use her laser beam eyes to drill the message into my head; it works well.
In general if one of my dogs comes and looks at me; they need something. It is then up to me to figure out what they need. This is not a leadership issue as some would say; but a way for my dogs to communicate to me. It may be that the water bowl is empty, they need to go outside, they want you to throw a toy, someone is in their bed, they want permission to do something or they just need to connect. Figuring out what they want is pretty easy if you know your dogs well and watch their follow up communications.
If it is a demanding communication, you should probably choose to ignore it. But you should figure out what it is that your dog wants; afterall they cannot say "I'm really thirsty and there is not water in our bowl." So watch carefully.