Is there a way to tell which members don't belong to a group?  I have a board where all members should be divided into one of four groups, and I want to make sure I didn't miss anybody.

No, there is not a way.  You should make a post group if you want to do this though, that way every member is added, among other reasons.

Would the post group make them able to post automatically?  The way this board works is that members sign up and they are either clients or from my company (or no group and no access if I have no clue who they are).  If they're clients they get access to boards a, b and c.  If they're from my company they get access to boards a, b, c, d, and others that clients don't need to see.  So I need to look at each registration individually and determine where they go, and thus, the opportunity for me to miss one from time to time.

Then this wouldn't work, you'd need to just keep check on everything.