LAN Connectivity Failure?

I have no problem with my internet connection, but I am unable to access my email accounts. When I run a diagnostic, I get "LAN Connectivity Failure: You are connected to a WiFi network that cannot communicate with our wireless router. Try restarting your wireless router, and if that doesn't work contact AppleCare customer service for further assistance." I have tried restarting the modem/router, which doesn't help. I have tried to have Cox refresh the modem/router, but I get a message that Cox cannot refresh it. I can access email on my iphone and ipad.

I have no tech expertise, so I don't understand how to fix this. I have tried online chat but it took so long I was eventually disconnected. I probably need a new modem/router, but I'm not sure what I need. I would prefer to buy something rather than rent. 

Can anyone explain this to me?