I recently got a Yoyager on a frame. The previous owner told me that 50wt 2-ply Aurifil cotton breaks, but 40wt Aurifil works with it. Also, after just watching a Superior Threads DVD all about threads, it seems that the higher speed machines need stronger thread.
I have been using a 60wt polyester thread to practice quilting, and it is working fine. I haven't tried the Connecting Threads poly yet, which is 70wt, but I am thinking it will probably be strong enough. The 60wt poly thread was one I bought at JoAnn fabrics; if you use a 50% off coupon, it would be a very reasonable thread to try.
In general, if you want the quilting to blend in then you want to use a finer weight thread -- in which case polyester is a good choice. If you want the quilting to show more, then a cotton thread like King Tut works well. King Tut is made with really good quality cotton in a manufacturing process that makes it very strong and very smooth. Having used Connecting Threads cotton years ago for piecing, I would not recommend it for quilting because it is very linty and may not be strong enough to stand up to the faster stitching speed of your Juki. For frame quilting with cotton, it helps a lot to use really good quality thread. From my limited experience so far, quality of polyester thread is less critical (plus the poly thread from CT is inexpensive compared to King Tut!).
It might be best to stick to cone threads. I think, for the Juki, you will need a cone thread stand. If you get one, be sure not to get the cheap plastic type, but rather get the heavyweight one. Or, Superior has come out with a new type of thread stand that looks really good too.