The
Petsmart /
PetCo cage will be fine. Use a different feeder, even a large jar lid will do the trick for that. With that deep a base on the cage, you may not want to use a standard bird waterer either unless you can make sure they can get their beaks in it good. For the chicks, once again use a jar lid, preferably with marbles to keep button quail chicks from drowning (they really do not swim well at ALL).
As chicks, you have to keep them warm and the cage won't really do the trick there. A box with either shelf liner or a towel on the bottom with a light hanging down works decently as a chick brooder. Do NOT put the water and food directly under the light tho. That can cause the water to heat up and if it gets too warm it can burn their throats or they may not drink it and wind up dying from dehydration.
Gamebird Starter Feed. For button chicks, you HAVE to ground it up some. It normally comes in too big a chunk and they won't be able to swallow it. A cheap coffee grinder works good. The same goes for most all baby quail regardless of breed. Purina is best. Using cooked egg all smashed up and complete with the shell is okay but not as the primary food. It should only be introduced going into the 2nd or 3rd week. The shell will help with the calcium they will need and the egg is protein.
When it comes to grit, you want to get the smallest grit you can. Check at a pet store. Many will carry a grit sized for finches. This works well for buttons. Pullet grit may be small but it's still a bit big for buttons.
Do a search here for quailladyoffortmyers as a poster and read everything she has posted in regards to buttons. Her information is based on solid experience and is one of the most knowledgeable people I have ever seen when it comes to buttons.
The rice thing: birds do not explode from eating rice. However, unless it is raw and unprocessed (meaning still has some husk on it) or fully cooked (where if you were going to eat it, you'd just slop some butter on it and dig in), I really do not recommend it. The reason being that they can fill up on the rice and it does take a bit longer for them to digest the white processed stuff than most other grains. That and it just looks like too big a grain for buttons anyway. Smaller grain like finch seed is better but only as a supplement and only after 2 weeks of age.