No one in checking with, regardless of Texas hometown or age had any idea this was a thing, either, and that's hitting people up to 60 years old and from Austin, Houston, Dallas, and San Antonio.
Maybe it's a rural Texas thing?
Not really. Maybe for the author. Living a long time in rural Texas (now urban), I can say that someone might have a propensity to use over and out for specific directions, but in my experience they are interchangeable and it's not a universal truth. Granted there are no universal truths about Texas/Texans so when you hear one, discount it. And you are right that up and down as cardinal directions doesn't feel uniquely Texan but can't tell when only on the inside.
In the seemingly endless rush to satisfy Texan exceptionalism, dialect and slang are among the traits that are always held up as unique. Yet every region has their own.
That never made any sense to me until now.
'Over' is more reserved when going to a previously known destination.
I went down (south) to Lubbock for College and over to Fort Worth (east) when I graduated. I still head over (west) to Lubbock occasionally because I have a sister going to Texas Tech. I go up (north) to Amarillo to visit my parents.
Maybe it's just me, but I haven't really noticed anyone using "out" to describe a direction. People do head "out" and go "out" all the time though.
Only within a modern western civilization context. In the ancient world maps were usually oriented to the East (towards the rising sun) rather than to the North. So, for them Up would have meant toward the East and down toward the West (where the sun set - usually associated with the land of the dead).