An Infected Box is a rare container with valuable contents. You have a small chance of obtaining it at a special location whose coordinates are indicated on a floppy disk.
You can get floppies with coordinates on the first floor of the bunker when you increase your specialist's reputation level upon filling its bar in exchange for identification tags, chevrons, and military notebooks. You can obtain these items in hard mode by activating the protocol in the bunker.
You can also get floppies in normal mode. You get a floppy at a certain reputation level in exchange for excess green, yellow, and red coupons from the officer in the same room.
When you get the Floppy with Coordinates, insert it in the terminal in the Overseer's Room on the first floor of Bunker Alfa. After that, a temporary location will appear on the global map. That's where you might find an Infected Box with quality loot: weapons, armor, rare resources, and components. The Infected Box's contents may change depending on your reputation level with the Overseer. By increasing your reputation, you're increasing your chance to get a rarer reward.
It is worth noting that in most cases, the location will contain a common military crate with resources. However, sometimes an Infected Box emerges there instead. You can't open a box like that instantly—it has to be brought to the home base and cleaned in the Acid Bath. To get the Infected Box to the base, you need to reach the location at the coordinates on a chopper and load it into the trunk. When you return to the base, you need to place the box in the Acid Bath. You also need to install an Air Filter into the bath and add Acid, which will kickstart the cleaning process. You'll be able to open the box and get your rewards only after the cleaning is complete.