Woodie Guthrie sings about The Old Chisolm Trail, an important overland cattle-driving trail when cattle ranches were popular, and the life of a cowboy was glamorized (the cattle trade was at its peak from 1867 until the early 1880s; The Old Chisholm Trail is a song that dates back to the 1870s). Cattle branding was a necessary part of ranching. When cattle driving, every cowboy carried his own personal brand book. They contained brands of local herds, reports of stolen cattle, rough maps of cattle drives and other trail information that was needed for ready reference.
Through the scribblings in a brand book, it was often possible for stray cattle to be returned to the rightful owner. When a strange brand turned up in a herd being sold, the owner—sometimes several counties away—would receive a check for steers he had never even missed.
The song references a lot of a cowboy's experiences on the 'trail' (bacon and beans most every day, etc.) Guthrie also tells of finding a stray in the herd (though why his boss would order it killed is a mystery to me; maybe just to make it rhyme with "skillet".
Brands were fairly complicated things; there was a Flying U and a KU, so... I'm guessing it might it be the brand on the cattle? Two U's or 2U. There are a number of ranches named 2U including one in Nevada. There was a UU ranch in Texas when the trail opened. There is even a UU Ranch Airport in Whitesboro, TX; there's a UUBarr (would look like UU-) Ranch as well. It's mostly a 'Dude' Ranch, though.
It could have been Abel Head Pierce's brand, a Texas cattle raiser (1834–1900) who, when he started, branded a B, then BB, then UU, and finally D.