All inflections of der Student except for the nominative singular are Studenten, but the possessive determiner gives away the correct number.
accusative singular: „Ich werde deinen Studenten einstellen.“
accusative plural: „Ich werde deine Studenten einstellen.“
dative singular: „Ich werde deinem Studenten begegnen.“
Note that begegnen is an intransitive verb that takes a dative object
dative plural: „Ich werde deinen Studenten begegnen.“
So it all revolves around whether the object of the verb is accusative or dative, since both of those forms have deinen as the inflection, one for singular and the other for plural. Aside from rote memorization is there any way to know or infer that the verb object should be accusative or dative. I was under the impression that all verbs of motion (where the natural English preposition "to" applies) were dative. And can it be said that since English derives in large part from German, that the verbs of motion would correspond quite closely in both languages? Or is there a mnemonic that German kids learn to remember which verbs are dative? In this particular case, the verb "employ" involves no such motion, and hence should take an accusative object, but begegnen (encounter/meet) makes has no motion involved and yet takes a dative object - which makes no sense.
The vast majority of verbs in German take a direct, accusative object. This article provides a list of exceptions. I guess there's no good way of distinguishing these, apart from memorization. For many of these verbs, the dative object is in some metaphorical way the receiver of some action, but this alone doesn't work as a good rule of thumb either. E.g.
„Ich tue dir weh.“ (wehtun) / but also: „Ich verletze dich.“ – “I hurt you”
If you use the preposition to in English for a receiver (in a more literal way than above), chances of it being an indirect object are very high. And even verbs that usually take a direct, accusative object can take an indirect, dative object – exclusively or not. The meaning would of course change:
„Ich schreibe einen Brief.“ – “I write a letter.”
„Ich schreibe meiner Mutter.“ – “I write to my mother.”
As far as I know, learning verb list is the best way to go – or you might at least check regularly, whether any of the words that you encounter on Duolingo appear on the list.
"Student" is one of the weak nouns: http://germanforenglishspeakers.com/nouns/weak-nouns-the-n-declension/
The accusative singular inflection, which you need for the direct object, is Studenten
nominative singular: „Ist dein Student vertraut mit meiner Arbeit?“ – “Is your student familiar with my work?”
accusative singular: „Ich werde deinen Studenten einstellen.“ – “I will hire your student.”
Could this technically also mean I will stop your student or I will halt your student?
I don't know any context in which the stop translation of einstellen would apply. This is rather reserved for actions or processes, e.g. the cease fire order „Feuer einstellen!“
einstellen also has an adjust meaning, but that wouldn't fit either.