Permanent residents may not file for spouses or fiances like citizens. They may only reside in the US if they have a "dual intent" visa like an H-1B. If they happened to be in the US on and F-1 or J-1, they could remain but would be unlikely to get a new one since these require non-immigrant intent.
However, anyone with a green card can file for a "reentry permit" which permits the individual to remain outside the US for up to two years without losing the green card for not living in the US.

