The first problem I see is that you set out to run a Vile campaign and now it appears you succeeded. In your definition a Vile campaign is where characters don't work well together and now you are surprised that, well, your characters won't work well together and are selfish. Sounds exactly like your campaign description to me.
A Vile campaign doesn't sound like the greatest idea either when the object is good vs. evil either. Vild sounds self serving and so to each person good and evil is relative as you mentioned before. You probably need a more defined enemy that each of your players has a vested interest in defeating.
Without a lot of details it is hard to give a good in character answer, as least from my perspective.
You could wield the DM magic wand and have a game meeting and make them come up with a back story that can explain why they should work together.
Come up with an enemy that has personally effected each of the characters. Stole their favorite magic weapon, destroyed their town, attacked the group and almost wasted them together.
Come up with a reason for each of them to fight together. Power of the crystal promises each of them their greatest desire making them want to do something. When they take the crystal as a group, the entire group is somehow marked for death from another group (assassins, theives, other adventure groups, evil dudes, whatever) so if they split up the are under great danger of death and they have a better chance staying together.
Basically it sounds that you are going to have to give them selfish reasons to stay together because of the Vile angle of it all.
Maybe others have better ideas, but this one is mine.
((You better let us know what you did after it is all done