You would have to make that a constitutional amendment, since it flies flat in the face of the Constitution. "The right to run for office shall not be abridged ... etc etc (point of the clause in question being that the only reason you can prevent someone from running is he doesn't have enough posts, or is not a registered voter)
Which is a good thing, because that way I can vote against it. 
Yeah, whatever that is you just said isn't in the Constitution.
You are referring to "Amendment" (I use that term loosely) II, Section 1, which states:
"No one shall exclude any person from an election or position on the basis of their political affiliation, whether it be in a federal or regional election. No one shall be excluded from any position due to their nationality, race, religion, sex, sexual affiliation, or age."
Hilariously there is no technical requirement to be a registered voter if you want to be President, though there is for Senator. There are post limits on both posts.
Also the recent amendment might just give the Senate authority over this issue:
"The Senate shall have power through appropriate legislation to determine rules for the conduct, proceedings and format of elections to the Presidency and the Senate unless otherwise specified in this Constitution."
That was an unintended consequence to be sure, but it seems to just about squeeze in.
