Many owners of Rainbow Six Lockdown have reported that their Xbox console freezes up when they attempt to connect to the Xbox Live service. This is a known issue and Ubisoft (the developer) is working on a patch to fix the issue. However, until the patch is released there is a method that may get you online to play. It seems a bit odd, but it normally works and will get you online in Rainbow Six Lockdown until the downloadable patch is released.
The issue seems to be gamertag specific, in other words, your gamertag will lock up while another gamertag on the same Xbox will not. What is happening is that the Xbox is locking up during auto sign-in, on sign in via the multiplayer option.
The Workaround Until Ubisoft Releases a Patch
If you have automatic sign in enabled, go to the main settings for the Xbox console and disable it. Alternatively, you can unplug your ethernet cord, but it's easier in most cases to change the auto sign in setting to NO. Now load the Rainbow Six Lockdown game into the Xbox, and begin a single player game. Pause the game and go to the community section (which will ask that you sign in, sign in from here). Now abort the single player game and go to the multiplayer section, where you can now check your friends list, create, or join a match.
This seems to work for most people, if this does not fix your problem, at least temporarily, your issue may be something we do not have information on (possible something as simple as a dirty disk).
Getting More Support
Xbox Live Customer Service and Technical Support can be reached at 1-800-4MY-XBOX, however, they don't seem to have information on how to resolve the above issue. Again, as stated above, your particular issue may be something different, so if the above fix does not work call them or visit www.xbox.com/support.
Note: This information is time sensitive, once a patch resolving this issue has been released we will note it here.