Backstabber is firing on a Firespray and is in the rear firing arc of the Firespray. Does Backstabber gain the extra attack die from his ability or not?

Strictly speaking, no. Backstabbers ability says he has to be out of the printed firing arcs on the target ship. The rules for the firespray do explicitly identify the rear arc AS a firing arc.

Confusingly, the falcon's rules state that it may fire outside of its printed arc, not that it has a 360 degree arc, meaning backstabbers ability DOES work when he is not in front of the falcon.

From the FAQ:

Q: When exactly can Backstabber trigger his ability?
A: Backstabber triggers his ability only if no portion of his base is inside any of the printed firing arcs on the defender’s ship token.

We had this come up a while ago as a rules question.

http://community.fantasyflightgames.com/index.php?/topic/82145-backstabber-vs-yt-and-firespray/

People tend to confuse this with the Falcon because of the turret ability.

The difference is that the Falcon's turret ability allows it to fire at targets "outside of its firing arc," where the Firespray actually has two fire arcs - one front and one rear. So a Backstabber shot from behind the Falcon would get his bonus - a shot directly behind a Firespray would not.

A shot at the side of either ship, would.