Well, it depends on the implementation, but Bernie Sanders has been on record numerous times as stating that his Medicare-for-All proposal would absolutely ban private insurance. Elizabeth Warren is another Democratic front-runner who indicated that she's on board with that, though her proposals didn't necessarily reflect it.
For what it's worth, the M4A _plan_ from Bernie's camp that I've seen didn't actually specify that it would eliminate private insurance, but he's also been keen on reminding us that he wrote it, and knows best what's in it.
So while it seems unlikely to me that political reality in America would prohibit banning private insurance, there are definitely those who espouse banning it for the greater good, so perhaps that fear isn't wholly unwarranted.