A Creature can possibly still use the Eagle Whistle, but it would depend on DM fiat with regard to realism
If a creature's breathing is unimpeded by the Paralysis and Stunned conditions, then they are still capable of performing the movements necessary to fulfill Eagle Whistle's requirements, at least per RAW.
But there are a number of contingent circumstances that could change things:
- A paralyzed creature (probably) can't move their arms. So if the Eagle Whistle isn't already in their mouth when they get paralyzed, they wouldn't be able to use it.
- Your specific scenario, a user who is already using the whistle, is probably fine for the user, since they probably already have the whistle in their mouth
- A paralyzed creature might still be able to breathe, but they might not be able to keep their mouth muscles contracted in the manner necessary to keep the whistle in their mouth. It might fall out or get jostled from their mouth if someone attacks them.
- Normal breathing might not be an issue for a paralyzed creature, but that doesn't necessarily imply that they're still capable of performing the diaphragmatic motions necessary to blow a whistle
- One could argue a character with a high Strength/Constitution score might be able to push past this specific issue, pending a successful saving throw
As DM, my ruling would depend on considering all those factors in aggregate, combined with the specific circumstances the character was facing. As an example, if causing a player to fall would instantly kill them, I'd consider being more permissive and allow the interaction to work, at least for a short duration.