No. Carbonite's own requirements say its personal backup supports current Windows and Mac systems, and Linux is not among them, so the subscription stops being useful once you switch.
Carbonite by Carbonite · Cloud storage & backup · No way to run it
Carbonite documents its personal backup as supporting current Windows and Mac operating systems in both 32
bit and 64 bit versions. Linux does not appear, and there is no client to install.
Treat this as a deadline rather than a comparison. Your Carbonite backup belongs to the Windows machine that
made it, and once that machine is wiped the copy in the cloud will eventually be removed. Restore everything
you might ever want onto an external drive while Windows still runs, then check the restored files actually
open before you go any further. That last step catches more problems than people expect.
On Linux the equivalent job splits in two. A backup app copies your files on a schedule to an external hard
drive, which is the part that protects you from a failed disk or a mistake. Cloud storage keeps a second copy
somewhere else, which is the part that protects you from fire and theft. Carbonite did both at once, so
expect to set up two things rather than one.
If you only ever want one of them, choose the external drive backup first. It is free, it is fast, and most
of the time it is what you actually need.
The offsite half of what Carbonite did. Stores the documents and photos you care about with a provider you pick.
Your files: No Carbonite import. Files are uploaded fresh.
This doesn't have to stop you
One stubborn program doesn't have to decide the whole question. You can keep Windows on the same PC and restart into it only when you need Carbonite. Everything else you do can move.