1) Be comfortable wet or dry.
2) Have good traction on land and on the boat.
3) Have enough support to hold them to your feet while scrambling on rocks or around the boat.
I've done 40 or 50 Grand Canyon river trips and I don't ever take sneakers. I take 2 pairs of sandals. Something like Tevas or Chacos. (One of my pairs I bought at Target and they have lasted 10 trips and are still going strong.) They have straps over the toes and around my ankle from behind my heel. I can tighten them so that I can climb rocks and hike in them. They are rubber so that I can get them wet, they still have good traction and are comfy.
You may want to bring sneakers. We have a "Shoe Bag" on the boat for you to keep them in if you want them during the day. Just remember the whole water everywhere thing. Some people like the support of dry sneakers for in camp at night. You'll pack those in your personal bag. Campsites are often on very fine sand. Footwear that traps the sand can get uncomfortable. Flip-flops are very workable for camp too.
Critically important --- break in your footwear before you get to the river. Even if you have had them for a long time, wear them and soften them up again before your trip. If you don't want to do this then bring socks to wear to protect all those blisters you will undoubtedly develop. Socks may look really geeky with sandals but at least you will be able to walk. Sometimes the sand causes even well broken-in shoes to rub your feet and cause problems. We have lots of band aids to take care of you!