Person has been jailed several times and has violated probation.
This goes to good moral character. By itself it should not be an issue, The multiple jail times concerns me more, depending on the charges. I would be most concerned if the green card is conditional and you still need to file the I-751.
Ther are two issues to be concerned about
1. Do NOT apply for naturalization without speaking to an immigation attorney who is convesant with the criminal penalites for the crimes
2. Do NOT travel outside the US without consulting an immigraion attorney as your return could trigger removal proceedings depending on the criminal record
Criminal Defense Attorney
In NJ a violation of a restraining order could he considered an aggravated felony that would trigger deportation. You need to speak with an attorney familiar with NJ criminal law and immigration law before filing any papers, traveling or resolving your NJ court matters. For a free consultation call me at 201-394-4676.
The short answer is yes. A green card does not protect an alien from deportation. Also, domestic violence & violating restraining orders means the person is a danger to the community. He/she may qualify for cancellation of removal, but its a trial process where the person will have to establish substantial equities in their favor. He needs a good deportation defense lawyer.