Don't know about the eggs, but the snail is most likely a small trochid. This family includes the Stomatella species and their relatives. It could also be a turbinid (you could tell by looking to see if it had an operculum, or trap door, made of shelly material), and a few species of abalone may look similar when they are that tiny. In any case, it will be an algae grazer and won't hurt anything in your tank. If you get a chance to catch it and take photos in a dish of water, preferably from a few angles including the aperture, I can probably narrow it down for you.
Cheers,
Don