Don't worry too much but monitor your pets every day. I kind of went through that after I had a chinchilla that did get sick and I had to put him to sleep. To make matters worse my new pet did have problems but it was very minor and treatable (coccidiosis that was treated with antibiotics and replacing the wood in the cage). They really are just like any other pet and they will thrive when you care for them.
The fact is the chances your chinchillas will get sick from stress is low unless you do something extreme like put their cage in a room full of screaming cats for days on end. (or a couple visits from an aggressive dog) They will probably will never get a fungal infection and if you never take them out of their cage they will still live to be a good age. It would defeat the purpose of having pets but it's true. You can feed them hay only too, not the best thing for them, but they'll survive. My biggest worry with my pet is fear of injury when he is outside the cage and ironically the only time I've ever had one injured was inside the cage and as bad as it was it healed on its own with basic first aid.
One of the things that helps me is that I actually keep a chart of my chinchilla's health. I try to write in it every day. I estimate how much he eats in the morning, weigh him on scale, check the consistency of his poop before I sweep it up, and even measure his water bottle level. In addition to that I have a wheel counter that gives me an idea of his activity level. I also look him over when I have him out. I note if there's anything unusual, if he's had watery eyes, or is unusually vocal or jumpy or behaviors I'm not familiar with, even if it's just a mental note. What I've found is that many things turn out to be normal and I've long forgotten that I thought there was a problem to begin with. If there is a pattern of something negative (like his weight is falling with his food consumption) then I still don't worry unless it goes on for several days because there will be inconsistencies and fluctuations.
Hanging around this board isn't always good for that either but keep in mind that our chinchilla board is where people come when they have problems. Think of the thousands of others out there that don't.