No, they didn't

Let's chalk this up to overstyling -- grabbing for the first style rule that looks handy, rather than one that fits the content at hand.

They said, 'What?!' follows the form given -- twice -- on page 304 of the 2012 AP Stylebook for a direct-quote clause that follows the attribution. (Franklin said, "A penny saved is a penny earned.") But that's not what "said" is doing in this hed. The Twitter users here are saying a lot of things, but "what?!" isn't any of them.
 
The hed is trying to represent an "echo" question, often used to clarify something you either didn't hear or didn't believe.* The form is clear when you substitute another verb for "say":

I saw the candidate eating the rat before the witches' sabbath.
You saw what?

If you need to show emphasis (you probably don't), you might want to tweak the typeface:

She said, "Just had my bag peed on by a bomb-sniffing dog."
She said what?

So unless anyone said "what?" you say "said what?" rather than "said, 'What?'" And don't say the stylebook says otherwise.

* More than you may ever want to know about echo questions can be found in Huddleston & Pullum's awesome Cambridge Grammar of the English Language.