There are many instances where の and が could both technically be used, but where one is more natural-sounding (or seems more relevant to the intent of the sentence) than the other. I'd think I'd say that this is probably one of those situations, though I'd never say it the second way, personally; I'd always use の.
The clause 「彼女がどんな困難からも逃げない」 is most certainly grammatically correct and natural, but when you change it to modify the word 「態度」, the structure of the phrase changes to be 「彼女のどんな困難からも逃げない態度」, because you're talking about her attitude (hence the 「彼女の」).
You can rewrite it like this to see how the modification works:
「の」 is really the natural particle to follow 「彼女」 here--if you replace it with 「が」 it sounds a bit strange, since a 「が」 clause doesn't seem like the best way to modify the noun 態度.
Hopefully that explains it well--perhaps someone with a bit more experience in linguistics could give a better reason why the 「の」 version sounds better :)