You might want to try the Hiroshima Tulip needles. Here is a prior thread where these are mentioned for hand quilting:
http://www.quiltingboard.com/main-f1...g-t273593.html
And here is a link to the Youtube video that explains how these needles are manufactured differently from others (they use surgical probe processes):
https://www.youtube.com/watch?v=_UGcNRS6sF0
Edit: What batting are you using? I would think the batting is more responsible for the squeaking than the fabric (unless perhaps if it is batik fabric).