This is an old problem, but damned if I can find the threads. In any case, it's not a WordPress problem; the issue seems to be with the viewer's computer. I have run into it DOZENS of times.
I clear the cache and do a restart. This always, and I mean ALWAYS fixes it, which proves to me that it's something to do with my damn creaky old PC. Only twice in history has YouTube gone down itself, and I know it's not down now.
ETA: I see rather that you've got the video, not the white box displaying. Ignore the above; hit the STOP button rather than the pause button, then navigate away from the page and back. Hit Play and let it load at its own pace for a wee bit. I've noticed sometimes it hangs up if I hit Pause too quickly. Once it's loaded a wee visible bit you can pause it. That should work; if not, report back.