Would this USB hub work to power 8 Pi 3's? If not... What are clean powering options for a cluster of 8?
A USB hub will likely not be enough to power 8 Pi's. The spec is for 500mA per port (USB 2) and 900mA (USB 3). I don't know of any hub that could supply 8 Amps (1 amp per port). All of the Pi cluster setups I have seen are using a multi device USB charger. I would suggest doing a google search for Pi cluster and looking at what they are using. One thing to note is since you are planning to use Pi 3s solutions using the Pi Zero, probably don't apply as they have lower power requirements.
I don't think that the device you linked to will do what you want. It's power supply is 5V 2A, which is not going to last very long trying to power 8x Raspberry Pi B's or B+s. According to this answer, you may be fine with Pi Zeros though.
You probably want to get something like this. There is a class of device like this that will provide full power to all USB ports, and that's what you want to look for. I have one of these, and I tested with a Pi B+ when I got it, and it will power the Pi B+. Presumably, it will power any other Pi as well. The only thing I found that it won't power are some older Windows Mobile phones, but that could be because the batteries are too old to take a charge.
I went ahead and purchased the hub I originally ask about and it is currently working and powering all 8 nodes. The hub allanonmage mentioned should work as well. Happy clustering!