How can I squeeze grain out of chickens more efficiently?

Invest some time learning Animal Kinship. The higher levels of this skill allow you to squeeze more grain from chickens, not to mention squeezing the same chicken multiple times. Animal Kinship 6 and 7 also raise the chance that you'll get bonus goodies and super harvests to a level where you see them regularly.

You can also use the two Fecundity powders to force all animals on a level to 'drop' a whole bunch of their product all at once - meat, eggs, butterfly milk. If you do it near chickens you will get like 4-6 units of grain per chicken, which you can just pick up without using any energy. Be careful though, because any other player nearby is free to pick up any of the bounty as well, so it's best to do it when you're on a level by yourself and there are lots of animals around.