Logged In Requirement

The original article from Nexon says "Can't reach 600 minutes? You'll also be rewarded for simply taking care of the egg. The more time you've incubated your egg, the better your prizes!," yet under Details it says that logging out does not affect the egg's progress. Are there less points earned for the egg (and thusly worse prizes given) if one doesn't stay logged in with the egg for the full 10 hours?

The egg will hatch regardless if you're logged in or not. That is the progress they are referring to. In order to obtain the best reward, you need to incubate it to 600/600 by being online for a cumulative 10 hours before the egg hatches, as well as sitting next to an expiring campfire and playing music. 600/600 is not required for hatching. Think of it as a reward quality meter.

In other words, if you logged in or 5 minutes and then didn't log on for the rest of the day, the egg will still hatch and you will still get a prize, albeit a crappy one.