Translation:Why don't you eat here?
No good: "Why are you not eating here?" ?
That's fine, it should be accepted.
Shouldnt it be eszeled/eszelsz or something?
No, the -el is the ending.
It's because "eszsz" is not hard to say :) so the -sz turns into -el after -sz.
"Why not eat here" is not okay?
It is no correct English grammar. English uses don't or doesn't to make present simple negative sentence.