> Clinton is definitely not eligible to serve as President
This has never been established. The 22nd Amendment prevents him from being elected President.
Subsections (a), (b), and (d) [every succession category] of this section shall apply only to such officers as are eligible to the office of President under the Constitution.
"In case of the removal of the President from office or of his death or resignation, the Vice President shall become President."
No caveats.
Edit:
There are a couple ways to become president, and only one of them is through an election. Article 2 talks about "eligibility", but says nothing on term limits. Amendment XII only says "eligible" again. Amendment XXII brings up term limits, but only says "elected", never "eligible".
There's a "but obviously..." reply to be made, but I see plenty of opportunity to play word games here.