I once attempted to start an urban legend that voles are immortal, but that never caught on. Anyway, no, crabs aren't immortal. They are, however, highly immoral, frequently indulging in pickpocketing, blackmail and "outing" people who aren't really gay at all.
They may indeed be immortal as they don't make any progress forward through life - they only shuffle sideways. In my book, anything that works at right angles to the rest of us should be treated with the utmost caution.