I'll have to watch this show and see what it's all about.
I think I would take the following:
1. Axe - Defense + build structure
2. Ferro Rod - Easy fire starter
3. 2 Qt pot - Water collection + able to boil water
4. Paracord - Numerous uses
5. Knife or Leatherman - Probably the Leatherman; self explanatory
6. Tarp - Keep dry / use for water collection / etc.
7. Fishing kit - Could probably do without, but hey why not. Hooks be good for stuff.
8. Canteen (metal - not plastic) - Another water collection and boil apparatus
9. Tarp - Keep dry again + if need to move from main camp can stay dry
10. Sleeping bag - Might as well be comfortable, but honestly might trade this out.. as with tarps, solid building of a structure and fire (oh and all the clothes they let you take); would be fine as far as warmth.