How to force doors to close?
A storage or companion cube can't be destroyed, the only thing that will happen is that force is put upon it. If you would use a crate or something that's actually destructable, then it would be destroyed. But since a storage/companion cube has no gibs, and isn't programmed to be destroyed, it can't. It's all in coding.
Maybe you should try to create trigger_teleports in the doors, make the doors their parents, and let them teleport any objects (not the player) they might hit.
"Duct Tape is the answer."
I think you can still Kill it with I/O, it will just disappear without any sort of sign or signal. You could always have an env_explosion parented to it, and have that trigger when it is destroyed, but that's just one idea.
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Or enable a trigger_portal_cleanser OnClose and disable it OnOpen, which will fizzle the box.
(I know you can disable the 'dissolve physics objects' of a portal cleanser. Can you JUST dissolve physics objects without clearing portals?)
Probably, try unchecking the "Clients" flag since that is what players are classified as.
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Wow, lots of good ideas.
Since it's a companion cube, I'd rather not destroy it. That's just a preference, though. In another part of my map if they drop the cube in some poison water I display a message and restore them to an autosave point. So I could do the same thing if I need to.
My preference would be to have the cube fly back into the room they started in as if the pressure from the doors forced it. But I want it to come back into the first room. If it flies into the second room and the player is in room 1, then that's as bad as the cube being destroyed since they need the cube to open the door (which initiates a timer to close the door). So I'll look into those options as people suggested (such as Hunter's idea). I'm quite the newbie with Hammer, though, so it might take me a while. 
Thanks for all your suggestions everyone.