I'm pretty wary of shipping chicks. I've never done it, personally, but there are people who do. What is the post office like where you live? I wouldn't trust the post office here with a box of bricks.
If they were out of water for half a day, that could have been the cause. With the heat, water is critical, and from your description of how thirsty the rest were, that is quite possible.
It is also possible that the chick simply failed to thrive. This does happen sometimes where there's...
I'm suspecting failure to thrive since you mentioned that it had to be helped out of the shell. Chicks that have trouble hatching usually have trouble for reasons.
I'll tag @007Sean here in case he has any insight since he has a lot more experience than I do on diseases, but I suspect this one...
That crackling sound isn't good. It certainly seems to be having trouble breathing in the video. The chick also looks under developed for 7 days if it's a coturnix quail. It should have pronounced wing feathers by this time and other feathers as well.
What is the brooder setup like...
It can be hard to trust Amazon reviews. There's a whole industry around paying for good reviews. With the new one, try boiled, crushed egg yolk and a bit of Nutri-Drench to see if that helps.
I wouldn't have the confidence to try that. It's so easy to injure something that small without meaning to.
You might try doing a couple more warm baths.
She needs companions. If you can't find other adults (exact age doesn't really matter once they're full size), you may want to see if someone who already has buttons will be willing to take her in.
I finally started to understand that when I started reading in other languages back in high school. In English, I don't consciously see the words on the page, but am rather hearing the dialogue, seeing the pictures. I take in entire paragraphs almost at once.
When I read Cyrano de Bergerac...
Did you get your bedding from a different source? I would double check the feed as well, especially at the bottom of the bag in case it got wet at some point. It wouldn't hurt to buy a new bag of feed just in case.
It isn't rare for one or two to fail to thrive, but there is certainly...
A warm epsom salt bath may help. With a chick it is very important to make sure it doesn't get chilled, so you'll want to make sure it gets right under the heat lamp when it's done.
Let us know how it goes!
I'm guessing that the chicks are not bunching up or spreading out and laying stretched out?
You should probably stop with the cayenne pepper. If they're getting a balanced diet, there is no need for anything else. They are also very small, and while small amounts won't cause any harm, too...
Pics? Can you upload a video to YouTube then post a link here?
What is the temperature at chick level on the warm side of your brooder?
The cool side?
What are you feeding them?
How many chicks are there?
What type of bedding are you using?