Connecting the Beaglebone Black to the KK Mini flight controller
The first thing to note is that if you are utilizing the 5V &amp; 3.3V power supplies on the Beagle Bone Black, I would suspect that you should be alright with using this configuration.

The reason why is best explained in the answer here: How to prevent applying power to I/O pins before BBB ready?

To quote: 


  If the sensor is supplied from the same voltage source as the microcontroller, generally there is no problem. It's not a matter of the board being "ready" so much as never applying voltage to an input that exceeds Vdd of the microcontroller. The actual specification is: -0.5V to IO supply voltage + 0.3 V


If you are not using the power supplies from the beaglebone, I would suggest doing so.

If you are using the power supplies from the beaglebone, I would suggest additional troubleshooting. 

Some ideas for additional troubleshooting:


Attempt the configuration with removing the +5V and 10k resistor. You
should expect the output to follow the input. This will tell you if
the +5V PSU is causing the issue.  
Attempt to use the configuration
without the high side connected to the flight controller.