In Justice League, all the members revive Superman using the MotherBox.
When Superman flies away, the MotherBox gets dropped in a parking lot beside the Kryptonian ship, where it was convenient for the villain to steal it.
Why didn't the Justice League members care about the MotherBox? Wasn't protecting it the reason why they revived superman?
Because Current priority.
They just resurrected a super powered being and they need to see if it worked or not. They didn't have any idea how much time it will take them to make sense out of Superman and they forget to lock Mother Box.
Their priority was to check if Superman didn't harm anything on his ways, they didn't expect that they will loose Mother Box on the way carrying it while fighting Superman doesn't sound like any great idea either.
There are two reasons for this.
They had just revived superman. After a long discussion and argument if it would be a good idea and if it would actually work, they were in the midst of finding out who was right. Superman could destroy the whole planet if he wanted to because the last being that was resurrected from that very place lead to so much destruction. They all knew that none among them could take on him alone and would have to work as a team to stop him. And when he did wake up it wasn't thanksgiving but more venge-giving. So taking care of the motherbox leaving him there destroying things would be a bad idea. Moreover, if they succeeded in pacifying superman then they could all work together in finding the lost motherbox anyway.
It made for a longer plot point and hence more slow-mo action and fan approval.
The other answers are good but don't seem to directly address one of your questions:
Wasn't protecting it the reason why they revived superman?
No, they revived Superman to defeat
Steppenwolf
because they were not strong enough (they all tried and failed, both individually and together). Even without the motherboxes in the picture, he could still defeat them in combat, and thus could potentially end all life on Earth, it would just take a long time.
Originally, they were protecting the motherbox (which they didn't fully understand1) to prevent it from being combined with the others, so that the Big Bad Thing didn't happen.
Once they had the idea to use it to revive Superman, the priority was to protect it/hold onto it long enough to use it for that, the hope being that a revived Superman would be strong enough to defeat the enemy.
With Superman revived, as others have already explained, their immediate concern was to make sure he wasn't a crazy monster, like Doomsday was. Plus the notion of resurrecting someone from the dead (or stasis, if you prefer that term) is a pretty insane thing; you'd want to stick around and check out the result for yourself. They also probably weren't expecting the motherbox to fly off like that.
Footnote1:
One exception to that is Victor Stone; as Victor, aka Cyborg, was essentially merged by his father with the motherbox in order to save his life, Victor/Cyborg shares its 'DNA'... for lack of a better term. Given enough time, he could theoretically understand it pretty well. But he is only recently Cyborgified in the movie, so he's still coming to terms with his capabilities.
Because the rest of the league was caught up in trying to contain the recently reborn Superman. They either momentarily forgot about the Motherbox, or they were simply unaware of its location.
