I have a couple other thoughts.
One, as far as I know, teams are not obligated to publicize the exact nature of an injury, nor are they obligated to comment on a player's eligibility other than to notify MLB when a player is placed on the IL. In fact, strategically speaking, it would be better for a team not to reveal whether a player is able to play or not. For example, would it have changed Carl Willis' managing yesterday if it had been publicized before the game that Buxton would not be available?
Two, HIPAA gives a player the right to keep details of his medical situation confidential, even from the team. What I don't know about this is whether it's included a player's contract that he waives that right.