I'm down to guesswork at this point.
If Log In isn't lit up, does Paragon Chat connect to the server when you first run it? The green bar at the bottom should say a few phrases, the most common being "Loading", then "Loading Sequencers", then "Cleaning Up" before filling entirely with green. If it doesn't fill up, stops, or is a color other than green, what does it do?
The other thought: if you're in a college/school environment, or behind a firewall (like work), Port 5222 is what is needed by XMPP. Paragon Chat uses XMPP to connect everyone together. If that is blocked, Paragon Chat can't reach a chat server. There's no real fix for that for school/work, it's their network. Mostly schools are trying to follow federal laws to keep kids off of porn even if port blocking is a ham-fisted method to accomplish it. Workplaces, it's the Principle of Least Privilege. You get what you need to do the job, nothing more. If the Internet was wide open, viruses, malware, and hacking would be a cakewalk. The only way around port blocking would be using a VPN or Tunnel, but that is against most Usage Policies and might get you in trouble. Not worth it.
If you're at home, talk to your ISP and see if they block the following:
5222 TCP XMPP client connection (RFC 6120) Official
5223 TCP XMPP client connection over SSL Unofficial
5269 TCP XMPP server connection (RFC 6120) Official
5298 TCP UDP XMPP JEP-0174: Link-Local Messaging / Official
XEP-0174: Serverless Messaging
8010 TCP XMPP File transfers Unofficial
(Source: Wikipedia)
Most ISPs aren't forward about blocking services to their customers, even on "unlimited Internet" plans. The best way to know is to ask directly. The most important port is 5222. If that port is blocked, it's a non-starter.