"Meine, nicht unsere."
So in undefined nouns like this circumstance, do most verbs just take on the feminine conjugation? Or is there a rule regarding this?
"verbs take on the feminine conjugation" ???
If you mistakenly wrote "verbs" instead of "adjectives," then the answer is no. The problem with examples like this is they give you one sentence that is completely severed from other contextual information. In this case, the object they are referring to is apparently feminine. If they were instead talking about "das Buch," then the sentence would read "Mein, nicht unser."
If you didn't mistake "verbs" for adjectives," then you should know that the conjugation of verbs in German does not rely on gender (thank god). Also there are no verbs in this sentence.