They do like you to be healthy, but if you have it in your backpack, I seriously doubt anyone will ask about it. Also, while the boots do have pretty good ankle support, I know during 1st BCT you will probably wearing athletic shoes quite a bit.That's actually tough question to answer. They really want you to show up healthy. If you do bring your brace, they will ask you if you need it. You need to answer honestly.
The Basic boots they issue you actually have very good ankle support, so you may not even need the brace while wearing those (which you would be doing in Jack's Valley while running the courses.)