I think I might have an explanation.
The confusion seems to be with the expression "sucul de portocale". Yes, I tend to see "orange juice" as one thing as well, but that is not quite the case in every language - in a grammatical sense of cause. Germanic languages like English (and German even more so) have a thing for combining words to create a new expression/word like "orange juice"/"Orangensaft". But Romanian does not do this. It rather sticks everything together than build an acutal new expression/word. The literal translation for "suc(ul) de portocale" is "juice of oranges" and not "orange juice" (which is the english expression).
Now, applying the rule for relative possesive pronouns, where the relative possesive pronoun (in this case "her") comes directly after the noun it is refering to (in this case it is refering to the juice) you get the sentence "Ea bea sucul ei de portocale". A literal translation would be "She drinks juice her of oranges" - but this does not work in English because the English language requires the possesive pronoun to be infront of the noun it refers to (juice in this case), therefor you get "She drinks her juice of oranges." But because "juice of oranges" is not the English expression, you have to change this as well for a good translation which leads to "She drinks her orange juice." in the end.
I am not a native speaker, but my practices and studies tells me this: You are correct with 'Ea bea' - She drinks / 'sucul ei' - her juice' / but 'de portocale' indicates what kind of juice. Rather than using the word 'from' instead it would be 'of oranges' which would be translated as 'orange juice.' So the whole sentence "Ea bea sucul ei de portocale" wold be translated as "She drinks her orange juice." I hope this helped more than than cause more confusion.
it doesn't really solve the issue to me... "of oranges" or "from oranges" doesn't change anything. The literal translation, by the way, would be "she drinks the juice of her of oranges".
The point is that intuitively sucul de portocale is one thing, it is "a whole". It's the [juice of oranges] that is hers. Why would you separate them?
The explanation maybe is that sucul de portocale ei would mean that she drinks the juice of her oranges and not her juice of oranges. Otherwise I can't really understand why you would put ei in that weird position.
Sorry that I have to correct you: " she drinks the juice of her of oranges" would be " ea bea sucul ei din portocalele ei" which is weird in both English and Romanian. The sentence " ea bea sucul ei de portocale" talks about a girl that drinks her juice made of oranges. The oranges don't belong to her, just the juice. A simple sentence could be " Ea bea sucul ei= she drinks her juice" only that the sentence specifies what kind of juice she's drinking.
I think the sentence you suggest will be translated as "She drinks the juice of her orange" (as unnatural it may sound in English) which implies whose orange the juice is made of. However, I can't be sure about that, so it would be better for a native speaker to answer you this.