What Rick said was very helpful. To eliminate red eye, you have to try to get your flash off the camera.
You should decide whether you want to go with smaller flashes and still be able to use them on location or you want to use larger studio flashes.
You can still take studio flashes on location with power packs.
Studio flashes offer a lot of power and faster recycle time compared to speedlights. With pets moving around so much and having to catch them at the right time, studio flashes definitely offer that ability to reduced the missed shots because the flash didn't recycle (fully charge itself) in time.
Larger light modifiers such as softboxes and umbrellas will be able to give you some interesting catch light to offer that nice kick to the eyes.
I used an umbrella set up with two Nikon SB800s for this shot: