Would it be possible to connect 4 or more PS3 Eyes to one USB3 switch connected to a USB3 port?

Are there preferred brands of USB3 switches that would allow this (intelligently divide up the available bandwidth)?

We have not tested this by ourselves, so cannot recommend anything. We only tested 2 cameras connected to a single USB3 port through the switch (on a laptop) - it was fine.
However, we were adviced by one of our users that he was able to connect 4 PS Eyes to a single USB3 PCIe controller (on a desktop PC). Though, not using a switch, just onboard USB ports. He successfully recorded on 50fps in such configuration (with 4 more cameras on different controllers).

I've been able to record 4 ports at the PS3's top speed into a single USB3 PCIe card - along with another 2 PS3's into 'normal' USB ports.

If the USB3 Hubs are actually switches (simultaneous streams), then it should work due to the tremendous bandwidth of USB3.

I'll call tech support at one of the better suppliers and see.

That also depends on the implementation of support for USB 2.0 devices in USB 3.0 controller. Often USB 3.0 does not handle USB 2.0 devices itself but redirects them to accompanying USB 2.0 controller (in the same chip). In that case, the combined bandwidth of USB 2.0 devices is still limited by USB 2.0 controller.

I spoke to IOGEAR Toll Free: 866-9-IOGEAR (866-946-4327) about their 4 port USB3 hub http://www.iogear.com/product/GUH374/

They said it should work (4 USB2 video feeds at once) and I may give them a try?

It's your decision, I cannot give you any guarantee :(