Horizon has a 'new' quail shipping box (basically a modified chick box), though I'm not sure if they are selling them yet. Either way, call your P.O. and make sure they accept lives and ask them what time to bring the birds in. My P.O.'s outgoing truck comes around 4 p.m., so I take birds to the P.O. around 3-3:30. Put something in the box for them to much on, preferably something 'wet' (like apple or orange) so they have some kind of liquid. The only way to ship live birds older than 24 hours is via Express Mail, which is very expensive. But if the buyer is willing to pay for this then you're good to go!
I don't know how many the quail shipping box would hold, but I'd say you could easily get 6-8 in the single N.E.S.T. The economy N.E.S.T. would hold at least 12 but probably 14-16. Just remember that each of the boxes has a weight limit for the total package, the single weighs 1 lb. 9 oz. by itself. So if you are lucky enough to have some 1-lb.'ers you won't be able to ship more than 7 I think. I'm pretty sure the weight of the single N.E.S.T. can't be over 9 lbs., though it might be 7, I'd have to check.
Also, avoid shipping in extreme conditions, not too cold or too hot. Early-mid fall (or late fall/early winter if you are in a warmer climate) or late winter/early spring are probably the best times to ship adult birds.