kicker cvr122 sub

that website is not working for me, but im guessing your getting the sub with the kicker box too. if so it will most likely be the dual 4 ohm sub wired in parallel to bring it to a 2 ohm load at the terminals. the zx400 will be fine if you wire it in parallel.

Since there is no 1 ohm power rating on their webpage, it leads me to believe that it is not 1 ohm stable. 

If they make that same woofer in a dual 4 ohm configuration, it would be more suitable for that amplifier.  Your only choice with that woofer is to wire it in series for a 4 ohm load.

yes, you should have gotten the cvr 124 rather than the 122 for the zx400. so pretty much the answer is yes, you need a more powerful amp that will do 400w @ 4 ohm. this way youre probably going to get an amp that will allow you to upgrade your if you want a more powerful sub.  the TMA T2-1000.1 made by jl audio comes to mind. or the T2-500.1

This way you can add another cvr122 later and wire it to be 2 ohm load. warning: this is going to be more expensive to get this  kind of amp. another option would be to try to swap the sub for a 4 ohm version and run the zx400 on it.  really depends on how much cash you want to spend and what you want out of your stereo