I have the problem with meaning of that sentence:
Don't just "Hey" me.
Is it mean, don't say Hey to me, because I don't won't it?
Or is it mean, don't say only Hey to me, but say something more?
Or this mean something else?
I read it a bit differently.
I think the original speaker intends to have a conversation, but is getting the hearer's attention by just shouting "Hey!".
The hearer means: Don't just 'hey!' me. At least say 'Hey, John' or 'Excuse me, John, can we talk', or something polite like that.
No one likes to be addressed as "Hey!".