Oh, that sounds entertaining! The Tough Mudder bills itself as the "toughest one-day event on the planet." Seven miles long and with 17 obstacles, it's sort of a Green Beret course for civilians. There's running, of course, but also wall-climbing, rope-traversing, water hazards, more. It's actually a lot of fun.
What you need is simply a good, sturdy trail-runner. Don't worry about waterproofness—you'll be up to your neck in water, so that's moot. Montrail's Sabino ($100) would be a good choice. They have all-synthetic uppers, forefoot plates for foot protection, and sticky and knobby outsoles for traction. La Sportiva's Wildcat ($100) comes in at the same price and has similar specs. Salomon's XA Pro 3D Ultra ($120) adds a little bit of technology to the midsole, with a "3D Advanced Chassis" made from nylon that helps add torsional stability. They're good shoes—Salomon's long involvement with adventure racing pays off.
You might consider something that's more of an approach shoe, as a little extra heft might come in handy. Oboz Firebrands ($124) are almost a low-top hiker. Very ruggedly built, with good foot protection and very grippy soles. Or try some Five Ten Camp Fours ($115), which have an external heel "cage" for extra support, and sticky Stealth Rubber outsoles. They'd work really well.
Anyway, time to start training! Need to set up a rope course and mud pond in your back yard.