They need to be fed unmedicated chick starter or non medicated gamefowl or turkey starter.
I really wish people would stop saying this. Whether to feed medicated or not is purely a matter of choice. I feed some of mine medicated feed and some non-medicated, and haven't seen any differences in either group.
The easiest way I have found to feed baby quail is to make a mash. If you put the feed into a blender, chances are you'll have some big pieces and some powder....either one is still not going to work.
Keep a bowl of moistened feed in the refridgerator. This way it won't spoil. Whenever you need some, stir it around and mash it up with a fork, every time you get some out. If you have a big enough supply in the fridge, eventually it will dry out, which is perfectly fine. The moistened feed will also cut down on feed waste, if it is somewhat damp, they can't scratch it around.
The #1 thing to worry about (other than the size of the feed) is the protein content. Chick starter is fine, but if it is less than 20-24% protein you'll need to supplement with ground-up cat food, boiled egg yolks, or some other method of increasing the protein. Don't feed anything but their normal food (except for cat food or egg yolks) for at least the first 2 weeks.
I've read on here that meal worms are a good supplement for protein, but then I've read online elsewhere that mealworms have a very low protein content, so I would say just to use them as treats, if at all.
I wouldn't keep baby button quail with chicken chicks, simply for the size factor. Even the smallest bantam chick is going to be at least 2-3 times bigger than a button quail chick. Quail chicks also require a higher temperature in the brooder, which could harm your chicken chicks.