geen is used when the noun should have "een" in front of it when it isn't negated. So, geen is the negated form of een. Sorta. Probably not, but still.
Ex: Is dat een limoen? Nee, dat is geen limoen. (Is that a lime? No, it is not a lime.)
Someone correct me if I'm wrong, though. That's how I understand the use of "geen".
In which case, it would be more accurate as "the girls do not eat any lime", right? More literally as "the girls eat no lime", but that's a bit more awkward in English. "The girls do not eat a lime" is not really something you'd say in English (maybe "the girls don't eat a lime, they eat two limes!").
The plural form of 'limoen' is 'limoenen'. The sentence states that it is a single lime.
Well, in English you'd say "the girls do not eat...", but there's another problem with your translation: In English, "limes", the fruit, are countable, so you can't say 'eat lime' (you need, e.g. 'eat a lime'). In fact, as an uncountable noun without the article 'a', lime doesn't refer to the fruit, but to this stuff:
(You can find translations of that page into other languages along the left-hand side)
I don't understand when the sentence is in continuous form and when in general. Like 'eten' is 'eating'. Which one is correct in translation, "de jongen eten rijst" - "the boys 'are eating'/'eat' rice. Sometimes 'are eating' is correct but 'eat' is also not wrong. So basically which one is correct?