Why did Square Enix stick with Sony?

[Reasonable]

I presume you mean why isn't FFXIII on 360 in Japan? I think that's simple:

1 - sales of jRPGs on 360 haven't been stellar in Japan no matter it had more than PS3, and the evidence suggests that the big jRPGs will sell more on PS3 than 360

2 - PS3 way outsells 360 in Japan

3 - 360 owns America over PS3

4 - 360 and PS3 have even split in Others

So, Square Enix actually took the option to have their cake and eat it in my opinion. They didn't totally fall out with Sony by going full multiplatform but it won't hurt them a jot.

They simply don't need FFXIII on 360 in Japan and they know it, and they may have got additional favours from Sony to remain region exclusive.

MS so wanted FFXIII in the West that SE are sure to have secured great funding to put in on 360 outside Japan without having to put it on 360 in Japan. They had the power in the negotiation, not MS

So SE will see huge sales on PS3 in Japan, loose a tiny fraction of 360 sales, and gain lot's of additional 360 sales in West as well as PS3 sales.

They are also sure to have nice incentives from both Sony and MS whether they are money or other business incentives.

I don't see how they lost or stuck by Sony either - they suited themselves and did so while seeming to give both Sony and MS an Olive branch each.

 

EDIT: if you're talking about FFXIV, which you might be too, then that's simple so far as has been released to the public.  SE want an MMO with charges using a scheme they've come up with.  Sony said okay, MS said not on Live.  SE so far seem to have stuck to their guns and told MS they're not going to adopt MS Live approach.  So right now I'd say FFXIV isn't so much exclusive to PS3 as not available on 360 due to MS/SE disaggrement on licencing on that platform - i.e. if MS/SE do sort it out the title will be on 360.