My daughter wanted to go in the military in 2008 and she was told you had to have a greencard and after you passed basic training and after a period of time possibly one year then you get your citizenship. We both heard it from the Army, the Navy, and Marines. As for crimes committed by non citizens why commit them in the first place.

http://www.uscis.gov/files/nativedocuments/M-618.pdf Pages 17 and 18

Consequences of Criminal Behavior for Permanent Residents
The United States is a law-abiding society. Permanent residents
in the United States must obey all laws. If you are a
permanent resident and engage in or are convicted of a
crime in the U.S., you could have serious problems.You
could be removed from the country, not allowed back into
the U.S. if you leave the country, and, in certain circumstances,
lose your eligibility for U.S. citizenship. Examples
of crimes that may affect your permanent resident status
include:
• A crime defined as an “aggravated felony,â€