Well, all was quiet last night.  Of course, the baby chickens were inside the house so nothing could bother them.  I am still not 100% sure the eggs vanishing and the cage being opened were connected.  Here is a theory I am starting to get together.  My neighbor that I have never trusted MIGHT take eggs from me.  He knows when I am home and when I am gone.  He has been off work for the past few months and is always home.  I just watched one of my hens laying an egg and another waiting in line for the nest box, so I am going to leave the eggs and the door unlocked and see if they are still there after I come home from picking up the kids this evening.  I really am not worried about losing some eggs, I just do NOT want ANYONE near my chickens without my permission. Obviously.
Now, the babies.  While it is slightly possible it was a furry type thing, human makes more sense.  Also, the door that was removed is on the back of the little house so if anyone looked out a window they would never see the person messing with it.  It would have been much easier to open the front or top, but also put them within sight.  That makes removing that door make sense.  I have no clue what the neighbor's ex husband who moved in feels about chickens but I do know this.  One of my other neighbors has a dog who roams the neighborhood.  He is generally well liked and even lays in my yard with my chickens climbing all over him.  The ex husband threw a fit when he moved in. Apparently this land used to all belong to him and he sold it to the developer who built all the houses.  He told my neighbor that he sold it with the understanding it would be almost like a HOA with strict rules and regulations on what would be done at the homes being built.  This makes no sense as we are out in the middle of nowhere.  He has made a few threats against the neighbor's dog from what I understand.  As I said, I am not sure of his feeling on chickens but I feel it could be hostile considering everything else.  Since he seems unfriendly too, I can see him trying to let the babies escape so I would get rid of my chickens.  He knows they were out there because he has seen me on the deck taking them out holding and talking to them.  
I am going to look into some of the aforementioned ways to know 100% sure what I am dealing with, but I do think at least one of the issues is human.  I have not completely ruled out the possibility of a snake on the egg loss.