Verizon's network type is CDMA, while the Nexus 7 seems to work on GSM networks only. Verizon staff should have known enough to tell you that immediately, instead of "trying and trying" to make it work. These are the network types and frequencies and frequencies supported by the Nexus 7:
There are two things you can do:
(1) I've heard that a recent software update allows you to use the tablet on Verizon's LTE network. Note that it will not work on Verizon's 3G networks, so it cannot fall back to 3G in areas lacking LTE coverage. You will have no connectivity where there is no LTE signal. Go to your tablet's settings menu and do a software update, then pop in the LTE SIM card and try connecting to the internet with WiFi disabled.
(2) You can return the tablet and buy one from Verizon, which may or may not work with both their LTE and 3G networks. I don't use Verizon so I can't say for sure. Amazon's return policy is very generous and you can print yourself a return label by going to Your Orders under the Your Account menu.
Hope this helps.