A green card holder, or LPR as we say in the biz, seeks admission every time he or she comes to the border or adjusts status. Where with citizens admission is guaranteed, LPRs have to show that they are admissible (through display of the green card) and are not excludable (by committing various crimes).
The fun part happens when an LPR commits an act that would make he or she excludable but not deportable. At that point the LPR could not apply for citizenship and or leave the country.
That being said, once an LPR proves that he or she is admissible, it is at his or her discretion to refuse to answer questions or consent to any searches.