According to Modern Brazilian Portuguese Grammar (John Whitlam, Routledge, 2011), "either...or" can also be used for "nem.....nem". Here's an example that's included in the book (page 106):
"Não tenho aula nem hoje nem amanhã." I don't have class either today or tomorrow.
The reason seems clear: "I have class neither today nor tomorrow" is not what most native speakers would say. It's not wrong, it just sounds needlessly formal and a bit archaic.
The same is true of a sentence like "I have neither my checkbook nor my credit card". It's correct, it's just not what we would usually say.
So based on this, I suggest that "She does not eat either chicken or fish" should be accepted.
Technically--if we're going to go into prescriptive grammar--no, because if "eats" is placed after "neither," it would also have to be placed after "nor": "She neither eats chicken nor eats fish." Realistically, there would be no misunderstandings if you didn't do that.
What is the reason for the radical change between the phrase, for example, "The girl does not drink neither juice nor milk" and that phrase that I repeated, practically the verbiage "She does not eat neither chicken nor fish" and was not accepted ? Can they position me? Is it just a different way of forming the phrase? I think my answer should be accepted or at least have a reasonable explanation for it not to be.