Do bmx bikes with 1 brake have a 360 gyro?

Answer 1

BMX race bikes often have only one brake, at the rear. And for a race bike, there's no need to be able to spin the handlebar - so no gyro.

But of course, you could have one installed - if you wish.

Or you could set a bike up with only a front brake. In which case you don't need a gyro either - you just run the brake through the steerer tube.

Answer 2

No.

Only if you want both to do bar spins AND to be able to brake. Race bikes don't do bar spins, so they don't use gyros for instance.