Is that caused because I formatted the Hard Drive? Because after that the problem appeared, but I guess it doesn't make any sense. I was running Xexmenu from memory card from the very beginning.
And console actually sees a flash drive, I can format it in a Storage menu. I have Xbox 360 Slim 4GB Freeboot with 500GB Hard Drive. It's definitely not a Freeboot problem, because before, everything was fine.