I like NB sneakers, especially the 574's and the 1500's, so seeing them make some new basketball shoes would be pretty cool.
In terms of ankle stability, I have no clue though. It would be interesting to see any sort of study regarding the fact. The general trend has obviously been towards low-cut basketball shoes though, with the KD's, PG's and low-cut HyperDunks all being very popular these days.
I always felt as though that trend may contribute to ankle issues, but in my personal experience in low cut shoes it hasn't changed anything.
I'm sure that there is a difference, mainly due to the different technologies used by different brands, but at the end of the day I anticipate Hayward to be wearing a brace regardless of the shoe he wears, so it shouldn't matter (hopefully).