Rayon thread is notoriously weak thread and shouldn't be used for quilting. Laying the completed quilt on a bed and sitting on it is enough stress to break the thread.
The rayon in your bobbin isn't breaking when quilting because there is no where near the stress on it as is on the top thread but it can easily break once the quilt is completed and slightest bit of stress is put on the quilt, even the weight of washing it. I would strongly recommend you replace the bobbin thread with either cotton or polyester thread. It would be a shame to have all the work in a completed quilt only to have the quilting threads break from normal use.