Does this H-Bridge need any supporting circuitry?
At first glance, no. You should be all set. Some things to look for:


Threshold Voltage of the FETs - are 1-3 volts so a 5V control line will be fine. In addition, they are FETs, so you can directly connect. No need for any additional components.
Current - the h-bridge can tolerate currents in excess of an amp and your solenoid only needs 220mA - 350mA.
Drain-Source voltage - is 60 and the solenoid only operates at 6.
Enable lines - unlike the SN754410, the ZXMHC6A07N8 doesn't have them, so you are set there.


Pullup/down resistors at the gates are a good idea to make sure the gates default to a safe voltage when there is no output from the microcontroller. 

On the software side, just be careful to avoid shoot-through conditions. That is, when switching the FETs on and off, be sure there can't be a scenario when a P1 and N1 or P2 and N2 are both conducting, shorting the supply.