Wingspan - Player out of eggs
A player cannot choose to take an action that requires them to spend eggs they don't have.
This means that a player with no eggs cannot chose to place a bird in anything but the first column, they cannot choose to spend an egg to get an extra card, and they cannot take any bird actions that require spending eggs.
But bird actions and spending a resource for a bonus resource are both optional, so the only full action that is outright prevented is adding new birds.  A player will always have an option to take an action that does not require spending eggs.