the link is not the same. https://share.dhs.gov/common/help/en/support/meeting_test.htm is the link from the room sign in help option. Same response from the site if connecting using IE or Chrome.
Ok, so that is the first issue here, we are looking at two differnt server environments and versions. The first link you provided is to test access and functionality to one of the Hosted server clusters that Adobe maintains. The second is to a privately hosted Connect server, the DHS, and is an older version of Connect and on their servers. So the issue lies with the ability to connect to the DHS server, which is good that it narrows down the problem to that environment, not necessarily their computer.
When I hit the DHS meeting test page my results are:
Player Version: WIN 16,0,0,305
App-Server returned: code:ok, servers=rtmps://share2.dhs.gov:443/_rtmp://localhost:8506/
FMS Server connected with protocol: rtmps
Using TSL: no
BW Test Results: = _ latency=23 msec, down=2708.3 kbit/s, up=1592.3 kbit/s _
Add-in Installed
Add-in version:9,4,96,0
This shows that they are using port 443 (bold above) to deliver the meeting, so you'll want to look at port 443 and see if there is any issue with that or rtmps traffic being blocked. You may need to reach out to whoever is the support contact for the DHS deployment to see if there is something odd going on there if port 443 isn't the issue.