In my humble opinion, I believe that my Ubuntu VM (the DMZ web server) that is connected to my PFSense HP (router) does not have a path to the Internet. Adding the NAT adapter to both the router and the web server fixed this issue. I hope this makes sense.If you're doing this instead of adding the Bridged adapter, then it may be a good work-around, but it doesn't explain why adding the Bridged adapter killed ping.
PS: My supposed (external) zone is not really the Internet, it is just a simulation. I confused someone else earlier today, so I just want to clarify this.