I know that Jó means good and reggelt means morning but what is kívánok?
It means "I wish (you)". It is frequently added in Hungarian, not so much in English.
"reggel" is a subject and "reggelt" is a direct object. E.g. "this is a sunny morning" -> "ez egy napos reggel" since morning is a nominative subject here. Compare that to "I wish you a good morning" -> "jó reggelt kívánok" because "morning" has the role of the direct object in this sentence and it needs to be indicated by the "t" after the noun otherwise the sentence will be grammatically incorrect.
When I'm in Hungary, and people are walking up, and coming into the kitchen I have never heard a "kívánok" after the "Jó reggelt". Is this because we're family and less formal with each other? Or, can the "kívánok" be considered optional?