Sounds like a sprain. You can still front squat without having your hands/fingers under the bar. I would try to stay away from that position for a little while until it starts feeling better.
What you can do in the meantime is get some lifting straps, wrap both of them around the bar (depending on what kind of straps you're using, it's going to be like a slip knot or simply just putting the bar through the part where your wrist goes), set the bar on your shoulders and grab the long part of the strap while keeping your elbows up. Your position is going to be exactly the same except your hands will be just above the bar holding on to the straps instead of underneath.
It's kind of hard to explain so here's a picture: