It's not the Customs rules you should look at, but the air carrier you plan on using. I've worked for several US-based major carriers, and they all require an individual crate for each animal with exceptions for litters of tiny kittens/puppies/chicks/etc.
There are also specific requirements for the size and construction of the crate. Such as, they have to have a solid floor and be large enough for the animal to completely turn around. Don't just guess that the crate you usually take him/her to the vet in will be acceptable, it'll immensely help your stress levels on the day if you read as much as you can and prepare well before the actual moving process begins.