I understand this is a networking/firewall issue and not an issue with the Broken Link Checker plugin. I see it is using cURL to fetch the URLs so I SSH'd into my production machine and typed this:
curl: (7) couldn't connect to host
Alright time to talk to my network administrators. They said this:
"Generally, at most shared hosting companies the servers are directly on the internet. Here we put the serves on an internal private network and then NAT to the internet.
The curl command will have to connect directly to the private IP address of the server. In this case for XXXXXX that would be 10.5.1.66. So, the command would be as follows.
curl -H "Host: people-press.org" http://10.5.1.66/about/danielle-gewurz/"
I take it I'll need to hack the Broken Link Checker plugin to make it work with my unusual set-up. Unless anyone else has a better idea?