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.

I would just use made-up names, made obvious by using scarequotes on the first usage. Personally, I feel Alice and Bob is a little too generic (like a math problem).

I would probably choose two from Renee, Sophia, Marcos, and Hans. Names with some interesting oomph to them. I would keep the sex the same. Not because gender is a part of the story (well I doubt it is). But just you will have an easier time picturing the students in your mind and writing about them if you don't change the gender. Or possibly referring to them on the spot during an interview (no pronoun slips).

You could easily use alphabetically-sequential names, as we use in logic problems, communications, and so on. Typically this starts "Alice, Bob, Charlie", but you may wish to vary this for your paper if student gender is important.

The sequence of names often also uses "Eve" in communications and cryptography, as an "eavesdropper". It would probably be more appropriate to change this for your paper