You can use this unique proverb as is and it makes sense whether it is used by a human or a bird. In your story, the birds are personified anyway by giving them human speech.
Depending on your context, an option would be using a literal phrase:
Don't put all your resources in one place/thing.
Some online sources mention an earlier English proverb which expresses the same sentiment but it is not really used today:
Venture not all in one bottom.
Note: Bottom here means "a ship, boat, or other vessel" which is a historical sense of the word. You can still find some usages of this phrase online; thus, you can consider readopting it by possibly replacing "bottom" with "boat" or "ship" to make it more clear.
Additional thought: The analogy to pregnancy is not really relevant for birds as eggs are like an external asset. Thus, it wouldn't be strange for a personified bird to use this proverb. And perhaps, the human version could be "Don't put all your babies in one crib." in an alternate world.