If Kobun, grandjedi, and a bunch of people here are anything to go by, the picture begins to sort itself out.
Nintendo was very restrictive in the NES/SNES days. They charged high royalties, and they made life difficult in terms of giving publishers control over carts. Squaresoft apparently had a number of clashes with Nintendo on various issues in the SNES days, and they felt like they weren't being fairly treated. They were already feeling sour about the N64 after Nintendo pulled out of making CDs, so when Sony rolled around and offered them a better deal, they didn't waste any time.
Square was very unhappy with Nintendo, though, and they were riding high on finally being independent from that console manufacturer. According to Kohler's quote, they badmouthed Nintendo in the press, and according to others here, they helped convince some other developers (Enix most notably) to join the Sony camp. Yamauchi didn't like Square much anyways, so he didn't mind losing them, but the other stuff pushed things too far, and he was never one to take an insult lying down. Thus, Square was shut out of Nintendo properties until they started falling apart in 2002 and they had to humble themselves and ask forgiveness if they wanted to make GBA and Gamecube games.
Square probably didn't need to burn the bridges they did after agreeing to develop for the Playstation, but their decision to leave Nintendo behind was certainly logical. Nintendo's third-party practices were very difficult for the publishers, who had to buy expensive carts from Nintendo and eat big losses if they didn't sell out. Not only that, but CDs were cheaper, and held much more data, making for a much easier developing environment. It's not a coincidence that pretty much every third party stopped making exclusive games for Nintendo's console as soon as viable alternative platforms presented themselves.
Anyone who sees the Nintendo of this period as some poor wronged company betrayed by jealous third-parties has to be completely delusional. There was no reason for those companies to work in a less business-friendly environment when other alternatives were readily available.