From experience I know that when traveling in a Jeep CJ5 at 45 mph and an oncoming truck is also doing 45 mph and the truck drops a full bale of hay that hits your bumper, it is critical that your brush breaker be at least 3 inch pipe.
I have no doubt that a bale of hay hitting the Jeep windshield would have been a disaster.
I am reasonably sure that if the bale of hay had hit the ground immediately in front of the tires then that would also have been a disaster.
Though this is not a natural disaster situation, it could have been a tornado that dropped the bale of hay.
When driving across a 3 foot wide gap in a paved road, bottom depth unknown, a pair of sturdy planks comes in handy. Loma Prieta earthquake, San Jose, 1989.
Winch cable should be at least 300 feet long. An adequate supply of snatch blocks is very helpful.
Oxyacetylene torches, and the knowledge to use them.
Adequate tools to perform extensive repairs, and the knowledge. I have rebuilt a transmission and rear differential while camping, but I have never rebuilt the engine. Have also repaired leaf springs when the center bolt sheer off attempting to fight a forest fire. When rebuilding a differential it is very helpful to have a small can of white spray paint.
If all public roads are closed, then 4WD is a must.
If public water supplies are unusable then a backpacker style water pump/filter, and an accessible water source (creek, lake, etc.)
Siphon hose to acquire gasoline from somewhere.
Off road motorcycle so you can send someone out for aid when the road cannot be traveled in a 4wd truck.
Non-trivial first aid kit, and the knowledge to use it.
3 months supply of prescription medications. Might need more, depending on the disaster.
Chains for your tires. Even a 4WD truck with gnarly tread mud and snow tires may require chains in some snow/ice conditions.
GPS, maps, binoculars, compass, long range spotting scope.
Sleeping bags rated for at least 20 below.
Various sizes of tarps.
Ropes and bungee cords.
Axe and shovel. Whetting stone to resharpen the axe.
Water proof matches. Flint and striker for when the matches run out.
Duct tape and bailing wire.
Ham radio. In a really bad disaster you don't need a license, because the government will be shut down. In a pinch, CB with sideband and a bootleg amplifier might substitute for the ham radio. Dual whip antennas in a phase configuration will help get a signal out.
At various times I have had all of this equipment with my group. Not necessarily all at once.
If you have no experience boon docking or dry camping, then a copy of the DVD of my presentation a few years ago at the West Coast Rally about boon docking, and the means to play it.
If I think about it more I could probably add to the list, but this is probably adequate for the purpose of the original question.