As a kid I used to always hike in sneakers. When I started "really hiking" I started wearing really nice, full-leather hiking boots. Oddly enough, I found that I was rolling my ankles quite often. I assumed I must not have strong ankles and needed the boots.
After a quite few years of hiking in boots, I heard about people hiking in light weight running sneakers, so I decided to give it a try. I found it much better on my ankles. I think there were a few issues with the boots. They were heavier and stiffer, and my legs would get more worn out as I hiked. The stiffness of the boots meant that they wouldn't shape to the rock, but would instead roll. And because there was some small illusion of ankle support, I would be less careful with my ankles. I've been wearing light weight footwear for years now and it's been great.
My suggestion is to strengthen your ankles with ankle raises and other strength building exercises. When you're hiking, if you start to feel tired or a bit "out of control", slow down and take a break. Focus on careful foot placement as you walk. Also, consider trekking poles. I find that they take a lot of pressure off my legs, meaning my legs get less tired over the same distance.
Those are my thoughts. If you find that you still really need ankle support, you may want to consult with a doctor and get real ankle braces. Good luck and happy hiking!