No, it wouldn't turn us into a "papers please" police state. No, it wouldn't do anything that we haven't already done in a chaotic way already.
Like this case here of a passenger in a car being arrested for not showing ID at a traffic stop. Similarly, there are already stop and identify laws. I am not sure that a secure national id would do anything to make this situation worse and would be a nice way to get around issues like name conflicts in e-verify.