In the episode Extreme Pokémon, the day care man gave ash a (teal? blue?) egg in a glass case and he said “when the pokémon hatches, use the pokéball on top of the case to hatch it with.” So… is that what’s happening when you receive eggs that already have a pokéball from the day care man? (incidentally, was that the larvitar egg, or is it another pokémon?)

(I don’t know the dialogue from that episode offhand, but I think you mean to say “use the Pokéball to catch it with,” not “hatch it with,” because if Pokémon actually cannot hatch without a Pokéball then we have some serious problems here)

I suppose it must work something like that?  I mean, we can hatch eggs even if we have no Pokéballs in our inventory, and the baby Pokémon have Pokéballs automatically, so unless we envision Pokémon somehow hatching with Pokéballs, someone must be supplying free ones with every egg.  You can probably read into this, if you choose, all kinds of sinister things about being born into slavery (which could certainly be a very interesting way to take it), but I don’t think you have to for it to make sense.  If you think of the main functions of Pokéballs being protection and transport… well, no one wants the most vulnerable Pokémon on their team to be forced to walk everywhere and have nowhere to retreat to in case of danger or injury.  And the alternative – just releasing an infant Pokémon into a potentially hostile environment with no caregiver because you happened not to have any Pokéballs at the time – is clearly lunacy.  I mean, in practice we do that in the games all the time and in astonishing numbers, but you sort of have to give them points for trying…

(also I believe the egg you’re referring to is the one that eventually hatches into Ash’s Phanpy)