It's Texas. That isn't relevant. If it had been his own house, there wouldn't even be any discussion. Since it was the neighbors, there is at least a little room for debate, but he should get off.
In the State of Washington you can use deadly force to prevent a “forcible felony” which includes burglary. I’m not sure if the deadly force can be used after the bad guy has taken the stuff and is fleeing, but it would seem to me if you shot the guy running away you prevented the burglary. I need to check into that for my own sake.
Would you happen to have a link to the actual law dealing with this?