Yes. japanrailpass.net:

Most JR trains have Ordinary Cars (coach class) while many long-distance trains also offer Green Cars (superior class). With a Green-type JAPAN RAIL PASS you can use either class.

Note that there's only a Green Japan Rail Pass, not a "first class" one. This means you can use Green Cars (roughly, business class), but you cannot use the recently launched Gran Class (roughly, first class) cars without paying additional surcharges.

Yes. japanrailpass.net:

Most JR trains have Ordinary Cars (coach class) while many long-distance trains also offer Green Cars (superior class). With a Green-type JAPAN RAIL PASS you can use either class.

Note that there's only a Green Japan Rail Pass, not a "first class" one. This means you can use Green Cars (roughly, business class), but you cannot use the Gran Class (roughly, first class) cars without paying additional surcharges.