Why she comes out the day you notice
A sick hen in the flock is in two kinds of trouble: whatever is making her ill, and the flock itself, which will shove a weak bird away from feed and water and may peck at her. Separation gives her warmth, rest and food she can actually reach, and it takes a possibly infectious bird out of the shared air, water and pecking range of the others. Move her the day you see the hunched, fluffed, tail-down posture, the bird standing apart with her eyes half closed - hens decline fast, and a day of waiting rarely improves the picture for either her or the flock.
Setting up the hospital crate
A dog crate, a large box or a spare hutch all work, lined with clean towels or a layer of shavings, placed somewhere indoors and quiet - a garage, utility room or spare bathroom, away from the coop's airspace. She needs warmth without cooking: out of draughts, and if she is very weak, a wrapped warm water bottle at one end of the crate so she can move away from it. Set shallow dishes of water and food where she can reach them sitting down, and skip the perch - a hen who cannot stand steadily will only fall off one. Dim and quiet suits her; check her often, but let her sleep between checks.
Caring for her without carrying anything back
The order of your rounds is the quarantine: healthy flock first, sick hen last, every day, and wash your hands between the two rounds rather than after both. Keep her feeders and dishes in the sick bay and never swap them with the coop's, and wear the same designated shoes for her or clean them before walking back to the run. Once a day, note how she actually is - droppings, breathing, whether she is drinking and eating, whether an egg has come - because the trend across three days is what a vet will ask about, and what tells you whether she is turning the corner.
Return, or the point of the vet
Send her back only when she has been fully herself - eating, drinking, bright, normal droppings - for a couple of clear days, and reintroduce her watched, ideally at a calm point in the day, because the flock will re-sort its order around her. Some returning hens need a few supervised sessions before they hold their place again. Do not wait it out if she stops drinking, breathes with her beak open or with a rattle, fades hour by hour, or if a second hen shows the same signs - flock problems move fast, and that pattern needs a vet the same day. KinsLe's Pet Passport keeps those daily notes inside her own profile, so if you do call, you can describe exactly how she has been, day by day.