Do ChromeBook computer OS systems support BOINC?

No, but you can research how to install a hacked UEFI BIOS on a Chromebook to install Windows then install BOINC.

You can also run Linux in a container in ChromeOs. Just enable development mode and ChromeOS will install a version of Linux. I have been able to run BOINC logged in with the Science United account manager running in ChromeOS's built in Linux container. No side loading required; this is a feature in ChromeOS. You'll lose a CPU core however due to the overhead.

Or - dump your lousy bloated ChromeOS and load Linux on bare metal through the Chrultrabook project, which is what I eventually did to improve performance. The only real hacking you need to do is disconnect the battery to disable firmware write protection; everything else is explained step by step on the project website.