The United States already has a national system to identify its citizens and residents. It is called the Social Security system. While it is not flawless, it is effective. It is doubtful that a National Identification Card system would be any more effective in controlling illegal immigration than the Social Security Administration.
Green cards and visas should be sufficient to control illegal immigration. The last time an entire country implemented a national ID program was 1930s Germany under the watchful guise of Adolf Hitler. National identification programs shouldn't be necessary to control illegals--American companies should be doing their part by not hiring as many immigrants. There are plenty of Americans who need jobs, so start hiring them instead of foreigners!