Actually this isn't a weird idea at all. I actually play Eyes on uke all the time and it sounds great. I learned to play it on the uke before the guitar because it was easier and I love the song. The key is to play up the neck a little. Stay away from open chords in first position except for the A chord after the Bm at the end of the verse and the chorus if you want but I think the G, C and D barres sound better than open there too. Anywhere you can use a barre use it. It sounds closer to the original that way because of the way the chords are voiced.
Almost every time I play it I screw it up the first time and have to shift a fret but if I'm not mistaken (it would help if I had the uke in my hand rather than being at work) I play the Emaj7 as a barre chord on the 5th fret (barre the 5th fret and fret the 7th fret of the A string) then play the A barre chord on the 4th fret. Shift down towards the end of the verse for a Bm barre on the second fret and then the open A chord.
This chart should help if you don't know the uke barre chords.... http://www.ezfolk.com/uke/Tutorials/1fo ... forms.html
Enjoy...Eyes is a perfect song for the uke once you get the voicings correct.