I don't get it. Arizona has no jurisdiction over non-Arizonians.
Before reading further, you don't vote in a national election. The only actual national election is the vote cast by the Electoral College. When you vote for Senator or Representative, you are voting within your district or state to send a representative of your district or state to Congress. When you vote for president, you're voting within your state to elect Electoral College representative from your state who will then vote for President.
The states decide the rules for their elections (within certain restrictions of course). They absolutely can set eligibility requirements for a name to appear on the ballot. And given that what they're asking is pretty easily within the bounds of the requirement for the office itself, it's unlikely that this requirement can be successfully challenged (but I'm sure that wont stop some from trying).
Personally, I think it's a great idea. More states should follow suite. Since the federal government has shown no interest in closing this potential loophole, the states absolutely can and should step up to the plate and do it themselves.
But to respond more directly to your post. This doesn't affect voters outside of Arizona. What it means is that any candidate for president who hasn't provided those credentials cannot appear on the Arizona ballot (and presumably would also be barred from write-in as well). Meaning that if (for example) Obama doesn't provide this information to Arizona he automatically can't get any EC votes from that state. The sticky point is that as I pointed out earlier, you don't vote for the candidate. You vote for an EC member from that candidates party. All the Dems would have to do to circumvent this would be to put "Democratic Party Candidate" on their application and tell people to vote for the Democratic party candidate. The Arizona law presumably can't bar the Electoral College from voting for whomever they wish, so there is a loophole there I believe.
EDIT: Having thought about it a bit more (and read through the rest of the posts), they'd have to come up with an actual person to put on the ballot (since that person would have to present a birth certificate as well). As someone pointed out earlier, they could just put a proxy in place and have people vote for that person. The question is how much this would be seen as just more ridiculous twisting of the rules to avoid presenting a full legal document which most people would assume should be required as proof anyway.
It's a good approach because it doesn't specifically say "We're looking at you Obama!", but if every other candidate on the ballot in AZ (hah! Except maybe one other IIRC) files a full complete form birth certificate and appears by name there, and Obama doesn't, but he instead has some other person stand in for him, a whole lot of people will start to wonder wtf is up with that.
Edited, Apr 15th 2011 2:33pm by gbaji