I don't understand why schools think they can diagnose.
My 6yo has ADHD, is not medicated (although we're open to it if he needs it as he gets older) and (not to say your son does have it) he has had the same experience of functioning very well in one classroom and terribly with another teacher. It's truly shocking to see how environment/teacher can totally change a child's behavior. If you can change schools that might make a difference.
If you are able to, have him evaluated by a private behavioral psychologist or neuropsychologist. If they are able to do a full series of tests and evaluations, they can tell you if he has ADHD, but also what his strengths and weaknesses are, what types of things might trigger problems for him, and so on. We saw Dr. Michael Appleby at Loyola in Maywood (not in your neighborhood, I know.)
If you are in a public school, one thing to think about is that a diagnosis that shows your son needs accommodations to help him to behave/sit still/whatever won't force you to medicate him, but it may be able to force the school, through IEP process, to accommodate his needs. For example, to let him get up and move around a few times an hour, or whatever is helpful to him to stay focused. I would consider going through with the evaluation process through the school - there's no cost to you, and you aren't bound to any course of action like medication, unless you agree to it. But the school is bound to accommodate his needs by law.
This is a very stressful thing, I know. Good luck to both of you and I hope you find some solutions and answers soon.