I also don't intend to sound rude or apathetic, but if someone has stress issues with things that happen as a course of normal gameplay in most any game out there, they really should be reconsidering their choice of entertainment.
I second this. I'm sorry your friends have their anxiety disorders if that's truly the case. It's certainly unfortunate and I have no doubt it puts a big damper on their lives in Eorzea and out of it. But other people have legitimate medical conditions too. Should SE take all the lights out of WS animations and cutscenes because they can trigger seizures in a very very small minority of people? That's just as real a medical condition as the one you're concerned with and, I'd argue, the consequences of a seizure can be notably more severe.

I don't think it's reasonable to ask for a game to be built catering to every minority interest though. I wish people didn't have seizures. I wish they didn't have anxiety attacks. I wish lots of things like that. But the world is as it is, and /most/ people don't have those problems. The game is built for most people. People who're truly so sensitive to lights blinking or seeing someone's hp turn yellow probably should find other things to do.

And I'll also second, as has been noted, that most of ff14 is fairly stress free by most measures.