Strictly speaking, 'well' (the adverb) is more correct, but nowadays either can be used in this sense.
It's also worth noting that 'good' has a much wider meaning when used with 'feel'. 'I don't feel well' can only refer to your state of physical health. 'Good' can also describe your emotional state, especially in expressions like 'I feel good/don't feel good about ....', in the sense of happy, satisfied or confident.