Song with just A minor and C?

Other than Billie in the low ground(an instrumental), I cannot think of one. It would be so easy to teach.

Is there a popular song with just these two chords?

I can't think of any or find any. Quite a few with the tonic and dominant 7th chord though.

yeah lots of childrens songs are simply F - c7(or a different variation as UkeJenny stated)
which is not that much harder than c - am

Here is a link to several two and three chord songs.http://www.doctoruke.com/twoandthreechordsongs.pdf

Change that to A7 instead of Am and you have the beginnings of a whole lot of blues songs. There have been many old timey blues songs played with just one chord, probably originally played on a "diddley bow." Here's a previous thread that may help. Pick 3 complimentary chords and you have a 12-bar blues progression going:

http://ukuleleunderground.com/2009/06/uke-minutes-51-blues-ukulele-chords/

Look up songs in the key of G, that'll narrow it down to a few million!

I have no experience teaching ukulele and not that much teaching guitar, but that is what I used Billy in the Low ground for in the past. If I do give uke lessons, nothing could be faster and develop interest more than teaching a couple of one finger chords that enable a student to play a song. It is also great when showing how to improvise.

What about "The Red Tail Hawk"? It's Am and G.

Some of Johnny Cash's songs use both chords . .

I know tunes with Am and G ie Good old Rebel etc but I wanted to teach a someone a song with these two chords that can be done with one finger.

I do not remember a Cash song with only these chords?

Eleanor Rigby, F and Am. http://www.guitarforanyone.com/2_chord_songs.html

The change I-vi (C-Am) is pretty rare, I-V (C-G) leads the polls in two-chord songs. As said, Fsus2 (0010) combined with C7 (0001) is the easiest entry into chord changing, and you have a more universal I-V change with just one finger there. As for Am-C, only 'Psycho Killer' by the Talking Heads comes to my mind, although the chorus also passes by an F and a G chord and the bridge even pushes it to a Bm.

Why not stick to single chord? A lot of groove based music doesn't change chords, so look at blues and reggae. It does require the lyrics or rhythm to become more interesting, or else it will sound like practice.

- 'All around the kitchen' by Dan Zanes is basically an Am, but you can throw in a C to make it more interesting. You can play along with the youtube video this way.
- You can play Bob Marley's 'Get Up Stand Up' with the same single Am chord (the original is a Cm, that's not too far off, but the transposition means you can't jam along with Bob).
- Some say 'Papa was a Rolling Stone' is just an Em, but is actually switches back and forth to an Am (you could transpose that to Am and Dm, if you wish). I would start out by teaching an Em7 (0202) and call that the 'finger bridge'. When you're teaching, just start that song in Am and shout 'take me to the bridge!'
- Exactly the same for 'A Horse with no Name'

If you can add an F (I know, that requires two fingers already), the possibilities open up - if you're teaching a teenager, Avincii's 'Wake me up' comes to mind: Am-F-C-C throughout. The original is two steps higher, though, although Aloe Blacc does change keys when doing acoustic versions. You can even play or teach the song the abovementioned Fsus2 replacing the F, using three one-finger chords: 2000, 0010, 0003. Ideally, those use three different fingers: middle finger, index and ring finger.

Ooh, I got a good one for C and Am! "Stars on the Water" by Rodney Crowell. Transpose it to C and you're all set. Listen to it here (in key of A): https://www.youtube.com/watch?v=16iv0FnXMvs

Chords/lyrics here (in key of G): http://www.classic-country-song-lyrics.com/starsonthewaterlyricschords.html

I wish more people knew the song, but it doesn't take long for people to pick up on the chorus.