PuTTY connects to SecureShell.
Yes, it is probably the router/Internet connection.
It is rare, but some high-class routers have the ability to drop outgoing connections.
I'm not familiar with Verizon, but it is possible they have some kind of proxy through which you connect to the Internet and this proxy doesn't allow outgoing connection.
I had similar problem at work. The work proxy was blocking port 22 (standard port for SSH). I changed it to 443 which is the HTTPS port, and it allowed me to connect to my server. To do this change, you have to modify
/etc/ssh/sshd_config and change default port from 22 to 443 and then connect through that. But this requires you to connect to a server in the first place which is impossible as I see it. Or you could very quickly run to the nearest library, modify the SSH configuration file to listen to the HTTPS port instead of the standard port 22 and run back home.
But I think your best bet is to call Verizon customer support and ask them about it.