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.

'Shining eyes' can be used an expression to convey eagerness, attention to something, inpsiration, e.g. 'the children looked at their favourite teacher with shining eyes', or this quote from a review of a Ted talk by Ben Zander: 'Shining eyes are the looks of inspiration, respect and passion that surround a great leader.' You could also say, 'his/her eyes were shining with tears', as in someone crying or about to cry. Looks like 'bright eyes' has already been covered in the other answers.

I am familiar with the term "bright eyes" being used to describe eyes that are "full of life" (expressive) or considered beautiful. Shiny eyes is not one I hear used but I think of glazed over or moist eyes.

"Bright" usually refers to light. So by calling someones eyes bright, you can either be referring to how smart they are, how beautiful they are, or that they reflect light. That's all I can really think of.