I agree with Emily on this one... for the most part. Lots of hysteria over people's pets and I've seen this warning going around especially towards dogs. I have had to laugh at that one, the only time I've ever had a dog look up either a low flying bird or a loud plane. Only reason a dog might look up is to see what all the people are looking at, see "nothing" get bored and stop.
But... I would actually want to take some extra caution with birds on this one personally.
I have no idea if the moon s l o w l y moving in front of the sun is enough movement to make a bird suspicious but we all know how they can key in on things in the sky as an instinctive response to predatory birds. Would an eclipse trigger the same "what is that!?" response and cause them to watch it in self defense? No idea, probably not, but if I were anywhere remotely near the path of this thing I'd move a bird away from the window for a few minutes or supervise them to make sure they aren't watching.
No harm in being weird and overly paranoid in this instance to protect out little ones. Or at least I'll keep telling myself that since I'm a mamma bear.