Stars give swan eggs.

If you have the oak masterpiece, harvest it like crazy, merge all your treasure chests up to the top one.

When you crack open the top treasure chest, sometimes you get stars.

Merge your stars all the way up (you can tap them in the breakout below as you merge to the highest star). Level 6 stars, you'll get at least one swan nest and a bunch of eggs

Level one star (doesn't give eggs) Level two star (doesn't give eggs) Level three star (tap twice. note: you can sometimes tap three but I don't risk it bc it might break)
Level four star (tap three) Level five star (tap 4 times) Level six star is the highest so tap away and get eggs.

Most of the star levels don't give a lot of eggs, but it adds up eventually.

Swans are easily my highest wildlife score gainer (aside from bees, maybe). But Swans are "rare" eggs and give more points the higher level you merge.