Shoes are probably the most important piece of equipment or clothing that you'll take on the Classic. Your feet will give you more pain than any other part of your body during the race and you should take care of them as much as possible. You might think that you can ignore pain as well as the next bloke and you'll just suffer through it. That might tactic might work in a marathon or on a hike but it doesn't work on the Classic. The abuse your feet and ankles take is just too great. Even if you can mentally withstand the pain, your body can only physically take a certain amount of damage before it slows down.
Not very many racers take full-blown, ankle covering hiking boots. Some do but they are often the people who drop out. The terrain the Classic covers is rough and heavy boots might save your feet a little wear and tear but the extra weight is not worth it.
Many other racers take something smaller than large boots but larger than trail running shoes. These "day hikers" are not a bad choice if you're just looking to finish the race. They are still a bit heavy compared to what you could survive with.
Most competitive racers wear adventure racing or trail running shoes. Don't take regular running shoes. They're not built to withstand the rigors of cross-country travel and after 50 miles of abuse they will fall apart. And after they fall apart, your feet will fall apart.
Trail running shoes are stiffer than regular running shoes because of an insert in the sole and usually have more toe protection than regular street shoes. No matter what you wear, if you're running the race for the first time, your ankles and feet are going to suffer mightily.
By far the most common brand used by competitive racers is Salomon. Most equipment made for adventure racing won't work for the Classic because it's too heavy and won't hold up to the abuse. One notable exception is the Salomon Mens XA Pro 3D Ultra GTX
There's a reason why you'll rarely see these shoes for sale under $100. Salomon knows these shoes are the best and they are going to soak you for the maximum possible amount. If you look for these shoes, contrast the XA Pro with the XA Comp. When you try them on in the store, the Comp might seem more comfortable but the Pro is the better choice because it has a more rigid insole supporting your foot. You'll appreciate this when you're walking creek beds covered with large rocks like these racers on Cooper Creek on the Nabesna to McCarthy route.
The XA Pro's also come in a GTX version. This just means it has some sort of gortex sewn into it. This is just a marketing scam and will add unnecessary weight. Gortex or not your feet will be wet for the whole race anyway, so just get the regular version, if you can find it. It's not always available.
One more reason to get adventure racing shoes is the nifty little shoe-lace holder that most of them have. Shoe-laces are generally a problem when racing and if you get shoes with the snazzy "quick-cinch" shoe-lace system and a place to tuck in the laces so they don't get snagged you'll save yourself a lot of headache.
Obviously, the XA Pro's won't work for everyone and you should pick what works best for your feet. Whatever you end up getting, take out the paper-thin liner that always comes with shoes and replace it with Superfeet inserts. These will do wonders for your your feet over the long run.