Jax in this game (like in previous games) is just okay midscreen, but he is super dangerous in the corner. The problem with Hunker Down is that his punish usually consists of f33 1+3/Db1/Bf2 and it results in a knockdown and minimal corner carry, and it forces Jax to make another read on the opponents wake up. Grinnin Barrets punish nets Jax 25-35% and 3/4 screen corner carry, which allows him to go into his corner game much easier.
Hunker Down gives him access to better pressure with the extended strings and the command grab, but it results in more knockdowns and needing to make more reads as a result. The command grab is good, but since it is a high I don't think it adds to much to his pressure game. All of his tick throws can be poked through.
Jax's pressure game in Barret is not bad enough to warrant picking Hunker Down. His pressure is, for the most part, the same, but his damage and corner carry is significantly higher. I like both the variations, but I pick Barret if I wanna win.