Quails do seem to be more nocturnal than chickens. Mine are running around for some time after the hens have gone to roost, and they always all lay their eggs in the late afternoon or evening. I provide several of those plastic domes you can get for small mammal cages, and they use one of these (always the green one - I dont know whether this is colour, position or just habit) to lay in, but at night I think they just settle down out on the floor rather than roosting all cuddled up together as chickens do.
You could keep them in a chicken run for security but I think you would come to feel that they needed their own area or cage, with a dry floor covered with some kind of bedding that you could change regularly. As your run does get very wet, the original earth flooring wouldn't be suitable. Quails can stand reasonably low temperatures but are susceptible to damp conditions, so if you just wanted to keep them in a run you would probably need to clean out all the chicken mess, line the floor of their area with clean membrane, and then add a good layer of Smartbedz or whatever. They don't need a very large area- maybe work on 2 square feet each as a minimum - but it's hard to poo pick quails as their droppings are small, squidgy, and get mixed in with the litter. Also they are quite smelly, due to their high - protein diet. So you need to clean out and replace the litter once a week - another reason why an ordinary mud floor is unsuitable. Also if their floor isn't clean and dry they get mud balls on their claws, which need to be soaked off, which can be quite tricky. At the moment I have my 6 in a 4 X 4 ft rabbit run on the grass, so I can move it on every 2-3 days, by which time the grass is pretty filthy, but it does save a lot of cleaning out and I hose down the grass after moving it. no, it's not perfectly fox proof but not too bad round here, the grass is very good for them, and at the moment I can't manage cleaning out their indoor cages after an operation.
I think mine will be fine out of doors for the summer, though I have made a cover from clear plastic tarp to go over the cage and keep it dry, also to provide wind proofing in the awful weather we are getting and shade when it gets too hot ( maybe!) They definitely need dry shelter out in the open, although they are quite hardy they do look a bit miserable in the wind and especially the rain. I can't imagine leaving them out of doors once the autumn comes - then they will go back into their cosy indoor cages, where they have electric light to keep them laying when the daylight falls below 14 hours per day.
Quails are fascinating little birds, and mine lay an egg each every single day ATM, and they are easy to hatch and fascinating to watch as chicks. But you do have to be fully prepared to cull the surplus males. My last hatch was 7 boys and only one girl. When the boys reach puberty at around 6 weeks they suddenly start raping the poor girls and often inflict terrible injuries on them as they tear out clumps of feathers and skin from the girls heads as they hang on to their backs. The girls appear terrified of them and are constantly racing around trying to get away. In the end I culled all the males including those from my first batch, and now my 6 lovely girls live peaceful, happy lives together, no quarrelling, and produce daily eggs. If you are not fully prepared to do this, then quails are not for you unless you buy adults. More boys than girls will be hatched on average, nobody wants the surplus except as an exchange for breeding maybe, and even if you do keep one male per 4-5 females there will still be the others to dispose of. To get my six girls i had to hatch 16 chicks and cull 10 in the end. Luckily it's not as difficult as chickens - a pair of secateurs to the back of the neck does the job - and it's not too bad once you've got fed up with dreading to find a bloodbath each morning you go to see them.
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