If you are detained and questioned by police, you have a right to remain silent, and to speak with an attorney before and during any questioning. It is best to say “I want my attorney and I choose to remain silent” and then refuse to answer questions until you have a chance to talk to a lawyer.
It should be noted that failure to identify does not fall within your right to remain silent. You will be arrested if you don't ID yourself with name and DOB. They have to be able to figure out if you're wanted to conduct their investigation. But that's the most that is required and you should probably shut up after that.