How long can a sick chicken go without eating or drinking?

Most birds can go about 3-4 days without drinking and a week or so without eating. But you'll see a lot of deterioration during these periods prior to death. And mind that these are estimates and actual times will vary by individual.

It sounds like your flock probably has a contagious respiratory disease. See if you can find an avian or exotics vet anywhere within a reasonable drive - it's very critical that you have blood tests done to to determine what the exact issue is. There are literally dozens of different respiratory diseases, many of which present with identical symptoms but cannot be treated with the same medications. And absolutely check out your environment to be sure that's not the issue! Is it possible your birds somehow got to moldy feed? What kind of bedding do you use and is it being kept clean enough?

I've had quite a bit of success using a NutriDrench/ACV combo in my birds water (1 week out of each month or two) to treat Mycoplasma S. and G. I have no idea how it will interact with other diseases, but it sure can't harm them. On that note, being as Myco is the most common respiratory disease, if you absolutely can't find anybody to blood test your birds anywhere, then go online and order some Denagard. QC Supply is cheapest, 1 liter of 12.5% solution for $45 + shipping. It blows Galli/Duramycin out of the water in terms of effectiveness, and has saved the lives of many a bird in my flock. And while it is specifically recommended for use in Myco infections, it's not going to cause any adverse reactions if your flock is infected with something else.