The idea of “face” is very much the same as the idea of prestige or honor. Therefore, the idea of saving face for one’s country is similar to the idea of saving your country’s prestige or honor. It typically is done in a situation where your country faces some sort of a humiliation and you are trying to reduce the degree to which your country will be humiliated.
In the history of the United States, one of the main instances of needing to save face for the country was the ending of the Vietnam War. The US was clearly losing that war. It realized that it could not win and wanted to get out. But it wanted to do so without being humiliated. President Richard Nixon and his main foreign policy advisor, Henry Kissinger, had to try to get the US out of that war without making it look like the US was simply giving up. Therefore, Nixon embarked on a sustained bombing campaign against North Vietnam that was meant to force the North to negotiate. That way, the US could exit with a negotiated peace, not simply by giving up and leaving.
Thus, saving face for one’s country is making the best of a bad situation and getting out of such a situation without humiliating the country any more than is necessary.