I mean if pokemon get sent to your box this way, why not eggs
right? xD It's just inconvenient to look in the Tall Grass every 10
minutes and having to remove an egg from your full party everytime,
then putting it back afterwards because you don't want any of your
eggs to miss out on the EXP.
I know that this was not the intended use for the Egg Box and that
it's for when you accidentally adopt a wrong egg and don't want to
break your chain, but I use them to store extra Eggs when I have
too many and then I take them out when I have a day with less Eggs
to hatch.