Αη-Γιάννης (this is the correct spelling) means Saint John. Αη comes from the word Άγιος( = Saint).
Άει στο καλό means "I don't believe you" or "get lost!" Αει or άντε or άιντε means "go to".
Other similar expressions:
"'Aει χάσου", "'Αει στα κομμάτια", "Αει στο διάολο". All these phrases are swear words and they mean " get lost", "go to Hell".
For more details about άντε, check this answer:
https://learn-greek-online.com/ask-greek/188/please-what-is-the-meaning-of-ante
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Vasiliki Baskou, Instructor/Director, https://learn-greek-online.com.