The most robust way to do this (as heartless as it sounds) is to kill your animals and buy new eggs. The other answers here won't work if you need to move your items from say your ship to a planet, which it seems a lot of people (including myself) do.
I've found the best way to do this is by using bombs, but any method should work, just be careful of destroying your own stuff.
Then just purchase them from the outpost and place wherever you like.
If you don't have any way of killing them or have enough money or if you don't want to spend so much money just to move it, I recommend using admin commands to acquire the money to do so.
/admin will give you admin rights. Then
/spawnitem diamond x will give x amount of diamonds which you can sell for pixels. Then just purchase the eggs that you need. Also
/spawnitem bomb 1 in order to get a bomb to kill them.