Word press in HCnix - does running an auto update to newest version cause any problems?

Can we run a wordpress software upgrade after loading the click and install for wordpress without blowing the program up?

The wordpress program will not be blownup if you upgrade it. You can simply login to the wp admin interface and then upgrade the wordpress manually.
 
If you want to be sure that nothing goes wrong after the WP upgrade. Then prior to upgrading wordpress, you can take backup of your WordPress MySQL database and then also copy all the files related to wordpress stored on your webserver.
 
This will keep backup of  things before applying any update of the wordpress software on your system.

Appears to work just fine.