It is alright to include students' first names in a teaching statement?
I think you should avoid giving their full names, as opposed to just their first names unless they give permission for it. Better, using an alias for the students protects their privacy but doesn't cost you any thing in the telling of the stories. Alice and Bob, as usual. 

But some readers might actually wonder whether you were too personal with the students, not keeping a proper "academic distance". I often tell a story about two students who succeeded after failure. It is a great story, and it works without names. If they heard my story they would recognize themselves, I think. But no one else would. Not even their classmates at the time.