So you're somewhere. Your stomach starts to bother you. Your head throbs. And so you feel it's time to inform your current company of your physical well-being in case you should suddenly, y'know, throw up on their shoes or something. "Um, hey guys? I don't feel so good..."
And somebody looks down their nose at you -- while you're already feeling miserable no less -- and says, "You don't feel well."
"Wha? Huh? No, I have a stomachache. My head hurts. I--"
"The correct way to say it is 'I don't feel well.'"
OK, so thanks for that. Not only did I totally not need that at that exact point in time, but secondly, you're wrong. It's correct to say "I don't feel good."
Good describes your emotions, your mental position on something.
Well describes, objectively, how something is or how it was performed.
So think about it...when everything's awesome around you, you're happy, life is just smiles and rainbows...do you say "I feel well" or "I feel good?"
I knew that you would.
So when you add the word not, that means the opposite. If things aren't going your way, and you're feeling rotten, and your dog died, and everything just sucks...do you feel "not good" or "not well" then?