So the grammatically correct answer (not based on what may be said casually) is she and I. Ex: She and I went to the store. The general rule is that you look at the sentence with just I/me and no other pronouns. I went to the store- correct. Me went to the store- incorrect. So you would say "She and I went to the store".
Another example: She gave it to I/me and him. She gave it to I- incorrect. She gave it to me- correct. So it's "She gave it to me and him".
That is a "stressed pronoun". http://french.about.com/od/grammar/g/stressedpronoun.htm
"Her and I" is not grammatical English. Take out the "and I" and how would you say it? "She eats", not "Her eats". The English rule for "and me/I" is relatively simple like this - you use object form for both or the subject form for both, you don't mix them. The difficult part can be getting the object/subject right, though the only tricky one I know off the top of my head is "She is taller than I" - this is arguably grammatically the correct form but it's very common for people to say "She is taller than me". The reason for "than I" is if you consider it a shortened version of "than I am" (which is not universally considered right but hey, this is English - who said there were hard and fast rules!? :-).