Blew my Arduino. Did I wire this correctly?
It would be best if everything ran at the same voltage. (And it is necessary that everything uses the same ground.)  Which is possible as there are 5V Arduino boards.  Interestingly, the Arduino nano, according to this schematic, runs at 5V. 

In order to use the Arduino Nano at 5V, instead of using VIN / Pin 1, you should be able to skip the Arduino's 5V regulator and use +5V / Pin 4.  But this must be a regulated 5V power source.

Now, why are you having problems controlling the servo?

It is difficult to tell, but your boost switching power supply negative input may not be directly connected to the negative output.  With out a common ground, it may be difficult to create the proper signal to control the servo. 

To be specific, if the ground or negative side of the servo power supply is not at the same potential as the ground or negative side of the Arduino power supply, controlling the servo may be difficult at best and at worst you could burn something out on the servo or Arduino.

To check, measure with a volt meter to see if there is any difference between the boost switcher negative input and negative output.

To solve this problem consider using a USB power supply plugged into the Arduino Nano and powering the servo off the 5V at pin 4 of the Nano.  Or, keep using your 6V battery and 6V to 9V boost switching power supply.  Connect this as you are to VIN Pin 1.  The Arduino Nano's 5V power regulator should kick in and provide 5V at pin 4.