I'll eat my hat(*) if bog standard Linux or FreeBSD installations try to execute anything on a USB stick unless it's rebooted with that stick as boot medium
They won't do that intentionally. But bog standard Linux machines can certainly be infected just by inserting a compromised USB stick.
First of all, the stick will be mounted. Typically, this happens automatically, but if not, the user will still have to do it manually. The USB filesystem can be modified to contain just the right corrupt data structures to trigger a kernel bug, leading to a compromise of the machine. If you think this is far out, think again. This was 2006, but don't worry, the NSA has zero-days on file if they need them. It is well-known that kernel "oopses" (such as this bug in ext4 from 2013) can often be converted into full exploits by a sufficiently determined adversary.
Assuming your Linux distro has a graphical desktop, you may next try opening the stick in a file browser, such as Nautilus. (Or it may even autolaunch when you insert the stick.) This too can cause your computer to be compromised, if e.g. the stick contains a PDF, which has been modified to contain just the right corrupt data structures to trigger a userspace bug in the program that generates the PDF thumbnail. By the time you think, "Wait, I never put any PDF on this stick", you're already compromised. If you think this is far out, think again. This was 2011.
If you're really paranoid, you'll forgo filesystems and desktop environments entirely and just dd plain ASCII files directly to the USB block device. But if your networked computer has been infected, you can never be sure that it's only doing that...