Jim Jakosh.
Hi Jim,
Thank you.  Yes, Bougainvillea would grow very well in Arizona.  I also had never thought of the branches/stalks being of any use.  But when I saw the thickness and strength of this piece.  I was very pleasantly surprised. 
I don't think I will grow them again, I feel more inclined to use the space and trellis for something I can eat. Maybe Beans or Cucumbers. 
Regards .. Cliff.