Earlier is the series, it is shown that Morgan used to play quarterback until he got hurt. In this episode, he says, "I played a little strong safety back in college."
Terlingua is described as having a large population. In fact, the population was less than 300 as of the 2000 census, and the entire population of the county was less than 9000 as of 2008.
When the BAU team arrive, they're driving around in a typical black SUV. Terlingua is hours away from any major city. It's very doubtful that the federal government would keep an equipped vehicle on stand-by in such a remote area.
Mention is made of tunnels running under the border. There is no border fence in the Terlingua area (nearest would be in El Paso) and tunneling under the Rio Grande is impractical and unnecessary.
The sheriff's office is not located in Terlingua, it is instead in the county seat, Alpine, over 80 miles away.
At the beginning Agent Prentiss says, "Staging their heads in front of the police station suggests the unsub might be local. You would have to have a knowledge about how to do something like that without being seen." The only knowledge needed would be to know to drop the heads off when people are not around, like around 1 in the morning. This in no way suggests the perp has to be local.