Originally Squaresoft is a no name game company that nearly went under, but was saved by the wild success of Final Fantasy, a game where they just let their designers go all out and design what they personally wanted to play. This sets the tone for the company: designers pushing the craft of their games, mostly RPGs, in new and exciting ways. Not all of their projects are runaway successes, but they're all at the very least fairly interesting.
Eventually someone decides they need to make a movie and push the craft of movies forward too, and so they attach Final Fantasy's name to a totally CG movie way ahead of its time and develop all kinds of new technology in order to make it happen. It bombs and they lose 120 million dollars on it.
Hironobu Sakaguchi, creator of Final Fantasy and vice president of Squaresoft, voluntarily steps down, and is basically shamed out of the company. Squaresoft merges with the other RPG superpower in Japan, Enix, and becomes SquareEnix.
SquareEnix is now a company run by suits that hold a monopoly on the big name RPGs, and as creative talent bleeds out of the company it becomes something of a Japanese EA or Activision (a mega publisher). The first Final Fantasy published by SquareEnix is the first "direct" sequel in the series (X-2). The second one (XII) has a lot of its creative energy smothered by business decisions meant to maximize its potential audience and profits, with the creative lead having breakdowns from internal pressure and eventually leaving as a result.
Then, on top of the company having its creative core slowly eaten away by suits, this generation hits, and Japanese developers in general just cannot keep up with the changing dynamics of working on the 360/PS3. Despite how much money they've pumped into their games to keep them as visually progressive as they once were, a lot of their best titles end up on handhelds (TWEWY, Kingdom Hearts BBS/DDD, 4 Heroes of Light, etc), while their "HD" games take too long to come out, are plagued by compromises, or just end up in development hell.
Personally, I think they're going to be putting out some amazing games after all the turmoil and adjusting they had to deal with this generation, and I hope to see them empower new faces within the company to start some kind of JRPG renaissance, though it remains to be seen what will happen or if they really have gotten their shit together.