I tried the QUBE demo, which just left off in a room with nine light-up buttons after a short little segment. Is that the entire campaign: lighting up those nine buttons? Or is there a storyline and more puzzle-platforming to be had once you beat the intro areas, as there is in portal?
The square you describe is a panel of indicator lights that light up as you progress through the game as you suspected. The game has seven sectors, as indicated by the Steam achievements for the game, and each sector corresponds to one of those lights. The game does in fact end after completing all of the sectors, so lighting up the panel is in a sense the goal of the game, though it is more of a progress meter.
The game does have puzzle platforming and other puzzle elements throughout, but it barely has a story.
There is not really a story, but without ruining anything, the whole game is not just "lighting up those nine buttons." The game feels like Portal, in a lot of respects... but without the narrator who insults you.
Also important: throughout the game, until the very end, they constantly add new elements; so you never feel like you are just repeating permutations of the same old puzzles.
Unfortunately, the game is extremely short: I'm an avid puzzler, and I beat it in about two hours. Even those less puzzle-oriented should be able to beat it in 3~4 hours.