Putty connects to SecureShell.
Yes it is probably the router/internet connection.
It is rare but some high class routers have 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 internet and this proxy doesn't allow outgoing connection.
I had similar problem at work, work proxy was blocking port 22 (standard port for ssh) what I did is change it to 443 which is 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 than connect through that. But this requires you to connect to server in the first place which is impossible I see. Or you could very quickly run to a nearest library modify ssh config file to listen to https port instead of standard 22port and run back home.
But I think your best bet is to call Verizon customer support and ask them about it.