Yeah I know it's a bit early for a rules question, but I wanted to get opinions on this. After reading the rules I'm not sure how the Greatsword's ability works.

It reads: "Forced: After a unit enters this zone, The Greatswords gain one power until end of turn." Does that mean they gain the bonus when a unit played by the controlling player (of the Greatswords) enters the same zone as The Greatswords? Or does it mean they get the bonus whenever a unit (played by any player) enters the same zone (the corresponding zone owned by an opponent) period?

It doesn't refer to corresponding zone, so... it'd just be YOUR zone that the Greatswords are in. Seemingly only really useful on the offense (unless you've got tricks to put things in to play during opponents turn), during the Battlefeild phase IN the battlefeild (unless you have a way to get things in to play before the capitol phase).

'this zone' refers to your own zone. Since Empire has 'mobility' this is a good ability to have...

If it would count opposing units, it would mention Corresponding Zones:

Corresponding Zones

If a card refers to an opponent’s “corresponding” zones, it is referring to each opponent’s zone of the same name as the zone in which the card is located. If a card refers to each player’s corresponding zone, it is referring to the zone in which the card is located as well as each opponent’s zone of the same name.

Example: Tom has a Zhubar Engineers in his batlefield. The Zhufbar Engineers have the text: “ Forced: After this unit leaves play, each opponent must
sacrifice a unit in this corresponding zone.” When the Zhufbar Engineers leave play, their effect forces Tom’s opponent, Kris, to sacrifice a unit in his own battlefield.