I'm not a fan of holdouts in general, although I understand that players have little leverage in contract negotiations and have to use what they have. If you want to hold out to demand your market value then that's fine. If you want to feel undervalued and pissed off throughout the negotiation, then that's fine too. What isn't fine, and what I have absolutely no respect for on any level, is publicly airing dirty laundry to try to force someone else's hand in negotiations.
That's ********, plain and simple, and to stick to your own analogy I'm pretty sure John Wayne wouldn't have done that. He would've privately made damn sure the Patriots knew where he stood, he wouldn't have given an inch, and he would have held out until he got what he thought was warranted, but that would have been it. No crying to the media about how betrayed and undervalued and disrespected he felt. And yes, if the same thing had happened in reverse, I'd think less of the Patriots. But the reverse equivalence would be the team publicly insulting him and calling out the ways in which he hasn't lived up to their expectations, which of course they would never do because--among many other reasons--they're simply better than that.
The holdout isn't the issue for me. Players have their own interests to look out for, and a very short and uncertain career window in which to make a lifetime's worth of earnings from organizations that have no loyalty to them. So I get the whole 'mercenary' outlook, and will never begrudge a player for thinking that way, although I think taken to an extreme it can be pretty shortsighted and self-fulfilling. Mankins didn't choose to be a Patriot, and he didn't owe them anything. I think we're probably on the same page there, and even if I was otherwise inclined to hold that against him, the way he played through injuries would trump that anyway.
The sticking point for me is how he conducted himself during the holdout, which IMO was totally unacceptable because of how unprofessional it was. As a result I lost some respect for him as a person, although I still respect the hell out of him as a player (and for the most part as a person, since it takes a certain kind of guy to play through a torn ACL).