If you are talking about a person, 'bright' would be used rather than 'shiny'. Someone's eyes can be bright as in clear, alert, full of life. There is an expression 'bright-eyed and bushy tailed" which means alert and ready to do whatever is required (I don't know where the bushy tail part comes from, although obviously, people don't actually have tails!). I can only think that 'shiny eyes' would be used in a context such as advice on purchasing fish. Make sure the fish has shiny eyes, meaning moist and not clouded over (as opposed to dry, dull eyes) to ensure that it is fresh.