I don't know about quail.
But as regards chickens:
Depending on how many hens you want to have, you often can keep hens with 4 square feet of space each and no run. People in with long, snowy winters often end up with this for several months at a time, and it can be done year-round if you can keep them from overheating in the summer.
If one chicken egg equals 5 quail eggs, and you can get one chicken egg per 4 square feet but 5 quail eggs in 5 square feet, the chickens would be better.
But this does not work for very small flocks of chickens. Visualize a single hen in a space 2x2 feet, or 4 hens in a space 4x4 feet. They barely have room to move. But if you have 36 hens in a space 12x12 feet, even though they have the same 4 square feet each, they all have room to flap their wings, run across the room, dustbathe, and so forth. So 4 square feet each only works once the total space is a certain amount big.
If you want to give them more space than that, you still do not really need 14 square feet per chicken (4 of coop and 10 of run). Just make a "run" (10 square feet each) and in warmer climates you can skip the coop. Make sure the run has a roof, put the roosts at one end, and put partial walls to block wind from reaching the roosts. There is no need for an extra 4 square feet per bird of "coop" space in a setup like that. Or in cold climates, you can allow 10 square feet per bird of indoor space and figure it's an indoor run, again with no need for a separate coop. That works even for very small flocks. For example, 2 chickens in 20 square feet will be okay (although with 20 square feet, you can probably go up to about 3 or 4 chickens.)
Hmm, I wonder if the guideline should really be something like 10 square feet for the first chicken and 4 square feet for each chicken after that? I think that might take care of the small vs. large flock differences.