I go with Index finger on Eb. If I would use the thumb, I had to rotate the entire wrist clockwise, which would be more uncomfortable for me. And slower. Especially slower, if C follows.
Rotating the wrist is maybe okay if you are on higher octaves, but try it on a middle octave in front of your body, then it gets really uncomfy. At least for me.
Generell I try to avoid thumb on black keys. Not always possible, but here it is. Also think about to use it as a hand stretching exercise.
I am not sure how this felt to me in the beginning. I tried both options and both had pros and cons. But now I am fine with Index finger and it feels natural.
Hope this helps...
