There is NO singletrack moutain biking in Big Bend National Park. Bicycles are limited to the same roads as motor vehicles. This means you can ride the paved roads and the unpaved 'jeep' roads. Riding anywhere else is illegal. It should be noted that some of the unpaved 'roads' are pretty technical in themselves and will certainly give you a workout. Try hauling a trailer full of camping gear down the Old Ore Road, the Glenn Spring Road, or the Black Gap Road for some great epic rides.
If you really want technical singletrack, then you need to check out Big Bend STATE Park located west of the national park.