I doubt she has an infection, after only loosing a shoe twice. It's more likely there just wasn't a good place to put the nails and combined with thin walls, it just popped off.
If you want to go barefoot, talk to your farrier about it, look around for a barefoot trimmer as well and get them to look at your mare. It's not that I don't think she can't go barefoot,just that you want to have someone that knows how to trim it right or it will be all for naught. Also, a barefoot trimmer can tell you how quickly they think she can go comfortable and help you get some boots for the ineterim. The boots will cost about $120-$145 for a pair, but they will last a looooong time, and pay for themselves in saved shoeing bills. But I digress...
Going barefoot-when her shoe comes off, does she show ANY tenderness at all? If she can toss a shoe and a good piece of hoof with it and not miss a beat, that's a GREAT sign that going barefoot would take minimum adjusting for her. Thin walled horses just tend to need more frequent trims (4-6 weeks rather than 6-8) depending on the footing, amount of riding, etc as they will be the first to chip at the first sign of overgrowth. In a way, it's a blessing for them because they tend to break off in such a way to that is only cosmetically ugly, while tougher walled horses tend to get angles all off and then REALLY break off in dramatic ways if you let them go too long.
A supplement may/may not help. If it's because she's getting a poor diet, then yes, it probably will. Most of the time, though, the supps. don't really help. Ultimately genetics determine the thickness of the hoof wall, just as people have diff. fingernails (if I were a horse, I'd be a TB, cause my nails are thin and breakable, but my hubby would be a mustang-I swear he needs my nippers to cut his. LOL and I take vitamins, he doesn't). Also, once shoes come off, some horses will actually grow better hoof quality just from the improved circulation, but without added supplements.
However, if you want to try one, a hoof supplement won't hurt anything but your pocketbook .
I bet your mare could do fine barefoot and if nothing else, get some boots for back up in case you want to go down some rougher terrain. The hooves get conditioned to what they work on most, so if you always work on grass/dirt, that's about how tough her feet will be. But if you work her on rocks frequently, her feet many never need boots.