So the neigh card states you can only neigh a card that is played, so the debate is does the active effect of "You may bring a baby unicorn card from the nursery into your stable" count as playing a card?
No, this does not count as "playing" a card. From the rules:
Play vs. Bring Directly into Your Stable
"Play" specifically refers to playing a card from your hand during your Action phase or playing an Instant card at any time. If you "bring a card directly into your Stable," you do not "play" that card. This means that any card that is brought directly into your Stable cannot be Neigh'd.
Note that there are two print runs, and the earlier version included a different rulebook without this clarification.