At the moment it's not possible to access a repository with username/password, but we're hoping to have that included for the next release. Once that's complete, I don't see a reason why it wouldn't be possible to access a private repository.
For 1.Final, we're not planning to include any means of updating the contents of the GIT repository with a "pull", but it would certainly be possible to do so from the command line. Being able to "update" the application through GIT will likely be worked on post 1.Final
We now have the ability to connect to https and git repos using username/pwd or a passphrase, to unlock an ssh identity via REST with LIVEOAK-662
LIVEOAK-663 has been created to make this available through the Admin Console