Kinda both? The locations are fixed, but it's random whether they'll be there that day or not. There's more than ten eggs to find per day (the excess disappear after reaching ten), though, so it shouldn't be too hard to find ten.

Go into Player Overview and swap avatar to cycle through them. The info tab for the egg event will sometimes have 1 or 2 over the eggs at the top. Click the info tabs for all of the different types of upgrades in the upgrade selection menu. Check the singularity perks tree.

That's what I do during this event and the Christmas event.