Heh. First time parents, first time dog owners. You should say that in your original post when you have a question, and people will go much lighter on you.
My rule is as long as it's not sharp or likely to splinter, as long as it was small enough to fit down their throat without choking them, and as long as it's not long and likely to get knotted inside of the intestines -- so, things that would be an emergency would be cooked bones, which splinter, sharp metal that's pointed like a fork or a chunk of wire, it rules out long pieces of string or yarn because it will knot, and it rules out anything that they're unable to get in or out like large rocks and the remains of a nylabone -- it will pretty much, uh, pass, with minimal discomfort for the dog.
If your dog eats something fabric and the fabric or string is hanging out of the dog's butt when it's done pooping, snip it at the bunghole with a pair of scissors. Do NOT tug on it. It could be looped around a section of their intestine and tugging on it will tighten the loop and could cut their intestine or completely block it. Keep an eye on it, and if it doesn't come the rest of the way out within 24h, it's an emergency. (This goes for cats too. Snip, don't tug.)
What you need to do is monitor for stoppage or straining. So throwing up once or twice? That's a dog, maybe make them skip a meal. Throwing up bile for 24h straight? Yeah, that's an emergency. Throwing up blood? Definite emergency.
Diarrhea for a day? No big deal. Diarrhea for two-three days, any blood in the stool, or straining without actually being able to poop for a couple days? Emergency; go straight to 24h veterinary hospital, do not pass go...
Boxers have some interesting health challenges due to their breeding and physical form. If you haven't researched what to expect as the parent of a Boxer, please do. Please have your vet educate you about checking your dog for cancer and check at least once a month, monitoring any lumps you find more frequently. Also, learn a little bit about megaeso****us ... I know so many boxers owners who've had problems with that.
You do know where your nearest 24h vet hospital is, correct? If you don't, you should have it in your phone.