I don't think Square has a restriction on different IP same user. As long as you're logged out, you should be able to reload on a different IP with no problems. From my experience with Verizon Fios (I work in the IT field where we have dealt with Fios firewalling our ports and servers for NO reason), I have noticed they are more trigger happy on the firewall blocks than most other ISP's.

Make sure your have the ports Square has designated open and forwarded on your router. If you're still experiencing issues, you may want to give Verizon a call and have them whitelist the ports as they may just hammer them down on the node level.

Hope this helps!