Nancy - have you ever tried Birkenstocks? They come in widths, the sandals are adjustable and the Boston clog is a nice slip on. Mind you, I think they are ugly as sin (well, really, the sandals are pretty basic if you get the 2 strap Arizona or 3 strap Florida style. Pisa is a nice style - a little more coverage, but not very adjustable). I sold shoes for 12 yrs, and Birkenstocks were insane sellers year round. People love them and wear them til they fall off their feet. And the nurses loved the Boston clog style - and they pull some long shifts. I have a few pair I wear as slippers in the summer. They are horrible feeling at first - you need to break them in gradually by wearing them for an hour a day and increasing that as you go (so you'd want them for a few months before you go away to break them in). Then all of a sudden, you realize how good they feel. Awesome support. Not sure about the non-slip...but people wear them everywhere. Make sure you have someone fit you for them - they have to fit a specific way - your foot comfortably within the footbed - no toes hanging out our side of your foot spilling over. If they aren't fitting properly, they won't wear properly. If you wanted some with a little more 'style' - look for the Papillio line - same support, but you can get some funky patterns and styles, as opposed to basic and boring.
(for the most part - the Arizona style was the best seller - my quibble with the Florida - the straps were narrower, and if people buckled them too tight, the top strap would start to tear - well, we sold the Birkoflor material - a manmade material - if you got leather ones, probably wouldn't have that issue...and you aren't supposed to buckle them tightly either - your foot has to fall naturally within the footbed - and buckling them tightly doesn't allow your foot to properly shape the footbed - wow, after 4 yrs away from shoes, I still rem my Birkenstock do's and don'ts!)
As an edit - you can get Birks with soft footbeds (there is a little bit of foam on the footbed) for those who really hate the hard footbed. You wouldn't have as many styles to choose from tho.