Mike, the proper technique for turning at right angles on a scroll saw, is try to pivot the work piece on the back of the blade - you do this by pausing your cut, then applying slight sideways pressure on the blade, and smoothly turning the wood while transferring and keeping the pressure on the back of the blade. Then resume cutting forward in the new direction. If you are breaking blades while doing this, it sounds like you are trying to turn too fast. Some can do this movement very quickly, but that's just practice.
Just pause at the turn point, and by pivoting on the back of the blade, you will not be cutting anything but the tiny circle made by the blade width. Then resume cutting in the new direction. Eventually it will become one fluid movement, but concentrate first on doing 3 distinct movements, (pause, pivot, cut) without rushing it.