Can I use a regular bird cage for parakeet or finch? Or how about a bushel basket in the pic? (under the assumption that they don’t fly higher than 7 inches)
Button quail are prefectly able to fly - I've had one fly over a house. And they need a lot of space. They are also monogamous birds, and so you shouldn't keep more than two in an enclosure. 1x2 ft (around the usual small parakeet cage) can work as a temporary cage, but as a permanent solution for a pair I'd not go below 1½x2½ ft - and I would consider that small. 2x4 ft is a good size.