No.
Any disqualifiaction applies.
For instance, if the Speaker were a not a citizen, he could not be appointed....wait....
Read the Amendment.
It does not disqualify anyone from anything.
It says "no person...may be ELECTED more than twice".
Now, this presupposes the existence of Presidential elections, which are not required by the Constitution and which any State is free to do away with at any time.
Read broadly, I suppose, the Amendment could be a restriction on the Electoral College, since what they do is a sort of an election.
But however you read it, the affected persons are the ones who do the choosing, not the one who is chosen.
They could have easily amended Article II and Amendment XII by saying, "No person who has served two terms shall be further eligible to the office of President..., or, "Any person who has served two terms is subsequently disqualified from serving as President..", but they did not do that.
Amendment XXII only restricts becoming President through the mechanism of being elected.