In this instance, the correct good vs. well usage would be good, as saying you don't feel well suggests there is something wrong with your nervous system.
No, that's not correct. You would have a point if I were referring to my emotional state. I was referring to feeling physically ill, and in that case "well" is acceptable.
Examples:
I do not feel well today.
You do not look well.
The version I used was "I don't feel well," and it was in reference to a stomachache. Your criticism would only be valid if I were describing an emotional state with "well."