अपने is also correct, without context there is no way to know if there is one apple or many.
Select the missing word नेहा ___ सेब खाती है ।
I'd say this is because «her own» in English implies a strong emphasis (on posession) that is not present in the Hindi sentence.
It might help to think of अपना as «my/your/her/his/its/our/their own» to remember when it is applied. But it just replaces the 'normal' possessive pronoun (उसका in this case) when the possessor is the subject. So here it really just means the same as «her» in English.
And the other way around: If you want, you can even express the same emphasis in Hindi by using the literal translation of «her own»: «उसका अपना». (See also my answer to RaleighStarbuck)
So... I think I understand this, but just to confirm: अपना means "his/her" when the noun belongs to the subject of the verb while उसका means "his/her" when the noun belongs to someone who is not the subject? (For people who know Scandinavian languages, this is like the "sin/sit" versus "hans/hennes," yeah? So a Norwegian translation of नेहा अपना सेब खाती है would be "Neha spiser sit eple" and नेहा उसका सेब खाती है would be "Neha spiser hennes eple"). Does this also apply for 1st and 2nd persons? For example one must say मैं अपना सेब खाता हूँ for "I eat my apple" (I assume that मैं मेरा सेब खाता हूं is incorrect...) and "you eat your apple" would have to be तू अपना सेब खाती है (I assume तू तेरा सेब खाती है would be wrong). Is this right?
I don't know any Nordic languages, so I cannot comment on that part (though it sounds very plausible).
In any case you described the distinction between अपना and उसका correctly. As I am also still learning Hindi I checked in two grammar books, to be sure. So you don't have to take just my word for it :D
Here is the rule from :
«The possessive adjective apnā, 'one's own' is substituted for the possessive forms of personal pronouns when the subject of the sentence possesses the object. apnā agrees with the noun it qualifies.»
So this applies - no matter what is the person/number of the subject. In  there are examples given with हम and मैं, in  one with मैं.
Apparently where अपना can be applied, it also has to be. Even if there is no distinction to be made about who the possessor is, e.g. when मैं is the subject.
( explicitly gives two examples marked as wrong,  says the substitution always takes place.)
However, it is possible to use both together, for emphasis. So this is the exact same structure as English «my own». Example from :
« [ यह मेरी अपनी गाड़ी है ]
- ye mērī apnī gāṛī hai
- This is my own car.»
(The Hindi/Devanagari text in square brackets is my rendering of the Urdu sentence from the book.)
 Schmidt, Ruth Laila. Urdu. An essential grammar. London: Routledge, 1999. p. 24f, para. 213.
 Hälsig, Margot. Grammatischer Leitfaden des Hindi. Leipzig: Verlag VEB Enzyklopädie, 1967. p. 71, para. 182.
As a Swede I think you're right, but wondering the same regarding 1st and 2nd persons.
apne is used for "belonging to oneself". "apne kapde" => my own clothes.
uske: her / his. "uske kapde" => her / his clothes.