Any reasonably sophisticated jet (of which this is clearly one) will have access to more or less realtime weather data, as well as NOTAMs (notices to airmen) regarding closed runways and such.
I'm not familiar with the system's details, but I would assume that it would pick the nearest towered airport with emergency services that met the requirements of the landing performance. Landing autonomously on a strip in the middle of nowhere, with nobody around, isn't likely to be particularly useful in an emergency unless the aircraft is no longer airworthy - and even then, if you can keep it in the air, going somewhere big with emergency services is a better idea.
A towered airport will have air traffic control (not all small strips do), and if the jet sets the transponder code to indicate an emergency, and is making blind radio calls about intent as to what runway it's going to land on, ATC will then clear the airport for the incoming emergency, and keep everyone else out of the way - and roll the trucks to meet it once it stops. I doubt it will bother getting off the runway autonomously - just come to a safe stop, shut down, and then it's someone else's problem to taxi it off.
Aviation is somewhat nice in that way. Emergency traffic has priority. If I were to be flying a Cessna 152 along Lake Michigan and had an emergency with O'Hare being the nearest airport, I've got priority, and the oceanic heavies will get out of the way until it's resolved. Now, you may have some serious questions about what the problem was, but this is the sort of stuff resolved after the fact, on the ground.