I love my .17 HMR and if I were going to be going after fox my go-to gun would be my Marlin 917V .17 HMR. I might prefer a slightly lighter .17 for going after fox but there are not a lot of squirrels that get away after I get them in my sights.
I don't think it would leave an exit wound but do I think it would kill one very efficiently.
I prefer using the .17HMR on foxes myself. It's a lot more fur friendly than a .22 or .22WMR. The 17gr TNT load from CCI leave an indecernable pinhole going in and rarely exits other than a tiny fragment or two. Plus it seems to anchor them faster than a .22.
A .223 just leaves a mess.