Posting a video of he and his team mates in the locker room after a game really shouldn't be that big of a deal. They let the press into the locker rooms at times, so I'm not sure I see the difference. If there was nudity, profanity, or other unbecoming behavior due to the others not realizing they were being recorded or him posting it to hurt his team mates, then I would be concerned.
No, Antonio Brown shouldn't posted the video to Facebook because he should have been listening to the speech his coach was giving. He had no regard about what was going on with the team. He was just interested in getting the people on Facebook to pay even more attention to him. It was stupid to do the video and I hope knows that.
Antonio should not have posted the post-game video on Facebook. This is a scenario where the athlete did not give one ounce of thought to the situation or no respect to his coach and his team. He wasn't even listening to the coach and was more worried about his brand. What a schmuck.
No, Brown should not have posted this video on Facebook, because it's not Brown's job to be controversial or try and upset people. Brown should be more professional than that. Brown should have just let it go and should have just let people have their own opinions, instead of trying to be loud and in the news.