With my buttons, they usually are up and running in no time. The ones that I have had that were a bit slower usually did not make it. Is it possible the baby has some egg goo stuck to it? That could make it look wet. What type of flooring do you have for them? They seem to do better when they are put on shavings right from the start. I would keep it seperated until you know if it is going to make it, no sense letting the bigger coturnix beat it up.
Oh, and if the chick is sticky, that could make the eyes unable to open. Take a warm washcloth and gently wipe the baby off. But definitely keep the baby seperated for now.