Some things also depend on your router itself. Typing those two values into the DNS box is a good start. What's in the others? With mine, I need to provide a specific IP address, because, for whatever reason, DHCP doesn't work so well with my IP-provided router, so I have IPv4 set up to configure manually, have 192.168.1.100 (for example) in the IP Address box, have a "255.255.255.0" subnet mask, and have "192.168.1.1" for the router's IP. A lot depends on your setup. Have you any idea what your router's IP is (from your intranet, not the outside world)? They're usually of the 192.168.xxx.xxx flavor (in most popular routers).
OP hasn't said anything about a router, just that his MBP is connected directly to a westel 6100 modem.
I would speculate the Verizon tech assumed he was behind a router.
So, I'm thinking one of two things:
Its possible Verizon programs their modems to only talk to the MAC address of the CPU that was "registered" with Verizon when the service was initially installed. Which is the purpose of MAC address cloning in most routers.
or
OP, if you are just disconnecting machine one from your modem, then immediately connecting the MBP to your modem (without power cycling the modem) then the modem is still looking for/trying to talk to the other machine, and can't. (different MAC addresses)
So, if you don't have a router, disconnect machine 1, power off the modem, connect the MBP, power on the modem. Wait until the modem has finished its startup routine. Power on the MBP.
If the modem is not tied to a specific MAC address, and assuming no hardware failure(s), then that should take care of it, at least for the short term.
Long term, highly recommend a router if you are going to have more than one machine on-line.