Keeping my fingers going

That’s a tough one. Congratulations, and good for you. My middle child is a Marine. Boot was absolutely brutal. It was far and away the hardest few months of his life. On the other hand, my daughter’s fiance is in the middle of Navy boot right now (graduating at the end of the month in Chicago) and he’s breezing through it. I imagine the Army is about splitting the difference for those two – it will be hard, but not all-consuming. I don’t think you’re allowed to actually bring ANYTHING with you – at all – for boot. Neither of my examples could. NOTHING.
You could see if you could bring one of those Jamstick things, or even one of those squeezy fretbuilder exercisers – they’re small – but you might be told no to all of it, and just use air guitar and stretches. If you’re not sure what kind of stretching to do for guitar, I can find a video – they’re quite good and helpful. When you get OLD, you’ll have to do them if you want to play worth a shit at all.
Good luck and kick ass!