I could be wrong, but I think the author of this article, Zack Kelberman, got it wrong. I have never seen anything about a no trade clause in his contract. Not saying that there isn't, but I think if there had been, that would be addressed by the media, before even talking about trades.
I think there has been talk about no team will trade for Rodgers, unless they have a new contract on the table with him, which he would have to agree to. So to that extent, he has a "I will only agree to this trade if....." implied clause. So if the Packers wanted to trade him to a team he didn't want to play for, he couldn't stop it, but he could retire or sit out his last season. No team is going to want to invest draft capital in that situation.