No, nothing has been posted that indicates Byakugan can predict movement and it's even more of an unrealistic notion to say anyone can predict movement. If you mean guess, then sure, anyone could guess and then rightfully get smashed by an attack when they guess wrong.
If all shinobi could predict movement than the narutoverse would far more balanced and it would come down to who's jutsu trumps who's. Yes, some shinobi can predict some movement. This is usually presented as a skill level indicator, a highly experienced jounin or above will more than likely be able to correctly deduce a genin's fighting style and predict certain attacks. This is not the same as being able to read an attack like the Sharingan does in real time. No guess-work or experience needed, nor do you have to see the attack before hand. It's very, very different.
If Neji possessed such a skill he wouldn't have been trashed by the opponent's he's faced so far, like Naruto or the Sound Four ninja. Yet, he's been surprised by more attacks than he's actually countered. Honestly, the Byagukan has no advantage over the Sharingan in any of it's state's except the points already noted -- which, in a one on one fight don't mean much other than someone can't sneak behind you and you can use gentle fist correctly (should you know how to do so).
So the real argument comes down to the gentle fist abilities and not the Byagukan. I can see where this is confusing because you need Byagukan to use gentle fist correctly but they are in fact separate things. One is a fighting style with fighting techniques, the other is a Kekkei Genkai dojutsu. The thread was titled in a way that made me think this was suppose to be about two dojutsu and why one countered the other.
Edit: To be clear, gentle fist doesn't do you a whole lot of good against an opponent who is way above you, unless there are other factors involved. I'm not trying to be harsh, Neji is one of my favorites but you got to draw the line on bias somewhere, especially when comparing shinobi from way different tiers of skill and ability.