Bethesda's community manager Matt Grandstaff notes the last patch (1.3) for New Vegas has been reported to cause savegame issues for the Xbox 360, and advises players to decline the update. If you've already installed it, he notes a fix is incoming and a workaround available.
The update for English versions of the game should be going live later tonight. Exact time is TBA, but it'll probably be later in the night. For folks playing the game in other languages, the wait is going to be a little longer. When I have a better estimate of what that will be, I'll let you know.
For now, if your game save is working properly after getting the "missing content" notice, here's what we suggest. Load your game save after getting the message about missing content. Once you're in the game, make a new game save, and moving forward you shouldn't have any issues. Once the new title update is up, you can then go back to your older save files.
If your game is crashing after loading the "missing content" save, you'll want to wait until this new update goes live.
Again, I'll provide new details as I get them. Thanks for your patience.