You can make some rough calculations of the distance of the bird based on the angle of view of the moon and the angle of the bird silhouette. Here's an article on Wikipedia that explains how to calculate it:

http://en.wikipedia.org/wiki/Angular_diameter
The upshot is for small angles:

δ ≅ d/D

δ is the angle of the object in the sky

≅ is approximate equality

d is the radius of the object perpendicular to the angle of view

D is the distance to the object

(angles are measured in radians)

I did some rough calculations based on eyeballing the size of the shadow relative to to the apparent diameter of the moon and I estimated it to be about 1/120th the angular diameter. I'm probably way off and a much more accurate calculation could be done by counting pixels, but I'm too lazy to do that. The angular diameter of the moon is roughly 1/2 degree or about 0.009 radians. So my very rough estimate of the angular diameter of the bird is .00007

Solving the above for D and assuming we have a relatively large bird which is capable of obscuring about 1/2 meter, we solve the equation and get:

D ≅ .5/.00007 ≅ 6500 meters.

That seems pretty high to me, but I'm not familiar with how high birds fly during migration. I'm probably off in my estimate of the angle, or the bird may be smaller than my assumption. There's also some uncertainty in how closely the actual size of the bird is reflected by the size of the obscured area, since we are close to the resolution limit of the image. But as you can see, the math is pretty simple and could be done by anyone.

Edit: I admit to a high probability of calculating this incorrectly.

Edit again: A little googling seems to indicate that migrating birds do fly about that high on occasion, so maybe I'm not that far off on my calculations.

Edited formulae with unicode characters to make them more readable.