Jon has said that, but we know that he likes to alter stories about songs to fit certain occasions (Superman Tonight, These Days, All Hail The King etc.).
And I'm pretty sure that he's even told the Max Martin story in two or three different ways - ranging from "he was in the room" to "he was there and helped on the melody" or something like that.

Here's my take on the story:
- Jon hated boygroups and pop acts around that time, just like Richie did. There was a comment from Richie, pretty snide, around that period trashing young bands who had either been casted or not been making music for a long time. And who could forget Jon's arrogant line on the Zurich DVD "you spend too much time with those boy bands, you got a MAN band up here tonight baby!"
- Max Martin was THE essential hit maker around that time, firing in number one songs for the Backstreet Boys, N*Sync, Britney Spears and so on. But again, all pure pop acts.
- Until not long before the album's release, Say It Isn't So was reported to be the first single and there was not even a sign of It's My Life at the Webcast concert in February.
- I'm pretty sure the record company intervened since they didn't seem to have a smash hit at hand and they feared that Bon Jovi's comeback would tank. Which, let's face it, would've most likely been the case hadn't it been for It's My Life.
- Max Martin has always said that his true passion was rock music, so when he was asked to collaborate with Bon Jovi, he surely was more than happy to join in.
- I think he gave the band an almost finished song: beat, basic structure, melody etc. were all delivered to them. Of course, they surely added the talk box, the Tommy and Gina reference and the line "like Frankie said 'I did it my way!'", but other than that it was Martin's work. Simple reason: the song represents nothing else they did in that era (or in the years prior to that) and feels greatly unconnected to the rest of Crush. At the same time, it's strikingly similar to many hits from other bands in the few years prior mentioned above.
- Jon feared being linked to the then-current pop acts with Martin's name on the credit list, so they surely reached a (financial) settlement that allowed them not to list him and most likely made him tons of money as a countermove.