Does sequence always take precedence? Someone says "you're fired". You say "you can't fire me. I quit". Is the firing all that's true?

I would guess they say that stuff just to save face. But if we are fired why not own up to it?  I have been fired.   Some people must think if they are fired that is a strike against them or something in the minds of others.  Or in their own mind. 

I would never say that if someone wanted to fire me. 
I say "Fine, I will be getting paid unemployment by you tommorrow!"
Unemployment is harder to get if you quit.