Chao don't live forever and, eventually, all Chao will go back into their egg cocoons and disappear.
There is no way to prevent this, and, if your Chao doesn't leave behind an egg, no way to bring your Chao back.
However, if your Chao leaves behind an egg after it cocoons and disappears you can hatch the egg and get the
"Reincarnate" of your lost Chao. This Chao has the same name of the last one and slightly leans towards the same preferences of the last one (physical traits, personality) but otherwise it's like a new Chao.
Your Chao have to be extremely affectionate with your character in order for it to reincarnate. Some tips to really boost their affection are...
Be sure to keep your Chao well fed, but don't overfeed them.
Try to pet them or cuddle them when they trip, they really appreciate it.
Chao love walks, try to take your Chao on walks.
If a Chao can't swim, they appreciate it when you take them out of the water when they're struggling.
Never throw or hit your Chao, that alone can kill them if it happens too much.
Give your Chao lots of Chaos Drives and Animals.
Every once in a while cuddle your Chao (petting usually tells them you approve of their behavior, cuddling just gives them affection).
Take your Chao to classes every now and then.
Let them know you like what they're doing, pet them when they do good.
Besides that, just overall be nice to your Chao and they will probably want to stay forever.