"John is not well: he is feeling dizzy."
Translation:János nincs jól: szédül!
It depends on the context. You're right in that it often means a lack of something, eg "nincs pénz" means "there is no money". However, Hungarian words and phrases don't necessarily have a 1-to-1 translation to English. In this context, you can think of it as a lack of well-being.