Japanese working for US company in Japan

A Japanese man is engaged to an American woman. They both currently live in Japan. The Japanese man works for an American company's Japanese branch as a contract worker. The American company wants him to be a permanent worker after he gets married because it's easier for them to "hire an American" than hire a Japanese.

Is that even possible?

After the two marry, the Japanese man still needs to apply for a green card. If he gets the green card, will him and his wife be allowed to live and work outside of the US as long as it's for an American company?

That makes no sense.

Getting married to an American does not make you an American, so you will still be Japanese as far as employment is concerned.

Working for an American HQ`d company`s Japan subsidiary has zero bearing on a Green Card. To maintain the Green Card, you need to be physically present in the US for at least one 24 hour period each year. This said, if you travel to the US (Hawaii, maybe) for the purpose of meeting this requirement may still see the card pulled, since the purpose of the Green Card is to permit you to live permanently in the US - which you obviously would not be doing if you live and work in Japan. You can apply for an `extension` at the US embassy - good for a year (extending the US absence to 2-years). They take away the Green Card when they issue that extension - but if you return to live in the US permanently within that one year, it will be returned to you.