Can you be punished under the UCMJ for not telling your chain of command of your DUI?

That will not be an issue these days--local law enforcement WILL tell your chain of command for you. I just went through the punitive articles of the UCMJ...if they were going to charge you with something, Article 134 is about the only thing that covers "failure to notify" (although under Shrub Bush's devastation wreaked upon Article 134, where there are now about a hundred specific crimes in it, that may no longer be the case). However! Since the local authorities are guaranteed to tell the military, expect at minimum a General Officer Letter of Reprimand in your file.