This is an "ir + a + infinitive" construction used as short cut to the future tense. Voy a comer arroz mañana. I am going to eat rice tomorrow. The difference is the infinitive is a reflexive verb sentirse. So the way you make the sentence is Subject + negation + reflexive pronoun appropriate to the plural 3rd person (se) + ir form (van) + a + verb without pronoun. Alternatively, one could write Ellos no van a sentirse bien. You have two options with the same meaning.
As an amateur linguist, I want to get it right. Just because something is common doesn't make it correct. Well (and bien, as I understand it) describe a state of being when used this way. If I say I feel well, I am saying that overall I am in a positive state of being. If I say I feel good, I am saying that I feel, I don't know, noble or something.
vneil: If one feels "well" that means they can feel (touch) something successfully or easily or correctly. If one feels "good" that means they have a good attitude, they have good health, or just a feeling of happiness, etc. You can say "I am well" to mean "in good health", but NOT "I feel well".
All of the following are correct English and roughly equivalent: "I am well, I feel well, I am a well person." This refers to health.
"I feel good" is also correct but refers to state of mind. "How do you feel about your exam? -- I feel good about it." (Positive, but not necessarily rising to the level of "noble")