It's really hard to identify your specific problem without observing.
My first question would be ... have you FMQ'd successfully with this machine in the past?
If so, then my next thought would be .... are you tugging on the fabric? or moving it too quickly? as you FMQ, so that you are pulling the needle and thus causing it to snap.
However, having it break in three pieces takes quite a bit. I don't think I have ever broken one in three before.
I know how frustrating this can be, so walk away from the machine for a while to let yourself cool down and relax.
Then when you are ready, go back and start fresh .... take the broken needle out, your foot off, the spool and bobbin out and then put them all back in place again, checking with your manual to make sure that you are doing it correctly, and with the proper settings, right foot and appropriate needle for the thread. It is so easy to do some little thing during this process that can offset your results.
Sometimes the problem just seems to mysteriously solve itself when we go thru these steps.
No doubt for the next while, you will be hesitant and unsure. Try to be observant, so that if it happens again, you might have a better idea as to at what step of the process it is breaking ... for clues as to how to remedy the problem.