Most states have a Braking Performance Law. Federal Motor Vehicle Safety Standards have a braking performance law. Usually, the requirements are identical. Sometimes the state will allow 5 feet farther stopping distance.
I have been researching this issue for over 15 years, and I have found ONE state that specifically addresses braking on a towed vehicle. That is North Carolina (GS 20-124).
Here in Montana the Braking Performance Law is MCA 61-9-312. There is no other legal requirement for brakes on a towed vehicle, including breakaway brakes.
In Colorado, a quick google search reveals 8 CCR 1507-18. However, it appears that this law only applies to trailers with surge brakes. Colorado also has a law which defines exactly what a "trailer" is. I did not look at that definition.
For those who are concerned about "emergency situations", where a dog or kid or vehicle dashes in front of you, keep in mind that at 60 MPH you will travel nearly 200 feet before your foot hits the brake pedal! Brakes on the towed vehicle or not, it is unlikely that you will be able to stop in time to avoid an accident. Good situational awareness is definitely required!