First conditional: When you're gone, I'll miss you. Not correct. Try "If you (are to) go, I’ll miss you."
Zero conditional: When you're gone, I miss you. OK
Is ARE present simple in both phrases? It is. See bracketed version.
Because in the first conditional, it's strange to think that ARE is talking about the present when it's giving us the idea of future. If it is present simple, are there any other verb tense options when you want to give this idea of future from first conditional? Present continuous or perfect. You can even use ‘will’ in the if-clause in this conditional if it’s not an auxiliary and it means determination as in ‘If you will(~determine to) go, I’ll miss you.”