I'm going to use 'reed switches' for this project which should work good. What's a reed switch? One of their uses are for burglar alarms, which are activated by a magnet and are pretty reliable. The second bottom piece on six boxes will need to have these switches and I plan on putting a slot in them to hide them.
 

Here is one kind of Reed switch, and like I said before, you would find this type used next to a door or window for burglar alarms. These types of switches are very common and as you can see are not very big at only 2" long. However, they do make them much smaller. To use them is also very simple because they only require two wires.

One wire is always attached to the COM (common or neutral) and depending on what your doing, the other wire is connected to N.C. or N.O. (normally closed or normally open). Think light switch here and that's exactly how these work. I'll be wiring these switches in the N.O. configuration but when the magnet is close to this switch, it moves the contact inside to the opposite position. In my case, closing the contact and completing the circuit. Once all six switches are closed, the connection is made and that's when the magic happens in the escape room.
 

I also added 1/8" thick spacers that I removed before cutting the slot to add more clearance (arrows). The yellow clamp is my stopping point and gets adjusted depending on the placement of each switch.
 

Here you can see one of the reed switches attached to the bottom of the box. I drilled a 1/4" hole in the back and ran the wires through it. Side note: the wires are 10 feet long, which is more then enough but it's better to have more than enough. 
 

Here is the hardware that I'll be using for the doors and hinges, but the wooden dowels and larger screws will be used for the figurines. What's a figurine? These will be used in each box to represent each one of the 12 days of Christmas, and six of them will have the magnets in them.