Translation:These apples are for my squirrels.
Duolingo's pronunciation is generated by a computer program, which I don't personally fully understand. It might indeed pronounce words slightly differently in the flow of a sentence, compared to when you click on them individually. This might be intentional, but if anything more due to the fact that that's just what happens when speaking naturally, rather than any attempt to include different regional pronunciations in the same sentence (which would be weird).
I recommend using www.forvo.com to double-check any pronunciations you're sceptical of.
Isn't Eichhörnchen a neutral gender noun, as other nouns that end with "chen"?
It is indeed.
If so, "für mein" seems the correct answer, not "für meine".
If the speaker was talking about one squirrel that would be the case. As the speaker is talking about multiple squirrels, "für meine" is the way to go.