I guess it depends what you mean by support. I think there are in general two main differences between a low profile trail shoe and your standard hiking boot.
Firstly the boot has extra material which encloses some or all of your ankle. The amount of compression this gives makes it harder for your ankle to move as freely as it would if barefoot and the main effect of this is no different whatsoever to wearing a good ankle brace (perhaps someone can correct me here if I'm wrong though). To evaluate this part, all we need to do is ask ourselves is it better to wear an ankle brace while hiking or not? I think it is really up to the individuals particular circumstances.
I am of the opinion that while wearing a brace may weaken your ankle muscles over time, this is very unlikely to be a problem if you only hike every few weeks for a couple of days (probably this is most people). So for infrequent weekend trips I would say that the benefits of wearing an ankle brace (or boots for the same purpose) may outweigh the drawbacks. If, however, you hike every few days, or for weeks at a time, then the brace will surely start to have some negative effect over time.
Secondly, and in my opinion, more importantly, boots often have much thicker soles which are supposed to provide more cushioning thus reducing the stress per second your joints endure upon every foot strike. The unfortunate drawback to this cushioning and increased height from the ground is that stability in general is far reduced. There is a large reduction in your ability to sense what is underfoot in order to compensate and make adjustments to your movements as needed. When you do misstep on a loose rock, the thicker sole works against you. When your ankle begins to roll, it is REALLY going to roll and no amount of 'support' (aside from rigid ski boot style shoes) can help because your full body+pack weight will be acting to roll your ankle further onto the flat edge of the side of your thick boot sole. Once your foot goes past the angle of no return, a quick recovery from the mis step is also made more difficult as a result of the extra sole thickness which gets in the way of any last second adjustment.
My personal opinion is that ankle support is really not the main issue to consider when attempting to reduce the likelihood of severe ankle injury. Ankle support in one form or another will be beneficial to some, but not others. The main area which I think can help in reducing sprains etc. is to increase your overall stability while hiking, thus reducing the likelihood of a catastrophic sprain. I think that the main ways to achieve this are:
-carry less weight. The less you carry, the less cushioning you will require. This is BPL after all so I think we pretty much have that one covered ;)
-try to improve your walking style. I saw a specialist a couple of years ago who analysed my walking gait. The information he gave me helped greatly when it came to picking out shoes and also allowed me to change my walking style slightly
-use trekking poles. add a lot more stability especially going up and downhill
-shoes with thinner soles which allow greater levels of proprioception
Anyways that's enough out of me for now. lol