The NHS in Scotland isn't just a part of a 'UK NHS'. In fact there's no such thing as a UK NHS. Our health service has been independent since it began in 1947 (you probably knew that but the glooms from Better Together like to spread this myth about a UK-wide organisation). But that doesn't mean that our NHS is safe from the privatisation that's happening in England.
You see, the money we get from the UK government (that's our money they're giving us back of course) is linked to the money spent on NHS England. That spend is reducing because the English NHS is slowly being privatised (which means that people have to pay for some health services). And because our NHS budget is linked to theirs, then the effect of privatising the English NHS is to reduce the budget for the Scottish NHS, even though all our NHS services are still free.
That's why the only way to protect your NHS and stay true to the NHS's original ideals of free medical care is to control things ourselves through an independent Scottish government. Because what do you think is going to happen when a Tory or Labour government in England turns their privatising gaze on our NHS if we vote no?