We did our freeze branding with a head catch, but no squeeze. It would have been easier with a squeeze, but this worked just fine. The cattle didn't move around much after the initial touch. The iron doesn't hurt - they just move when they feel the push of the iron. You have to keep pretty firm pressure.
Either dry ice or liquid nitrogen will work. I think the dry ice method needs a few more seconds of pressure. We used liquid nitrogen. There are some good instructions online. Try here: