Unable to send mails on Mac after router installation

Mac user for years, first time this problem. Instigating factor seems to be internet provider provided a new router because old one wasn't operating properly. After rounds with internet provider and email server provider, they are stating it is a "mac problem." Hours on phone, reading forums, trying everything anyone suggested, my wife and I still can't send emails without restarting our laptops/ipads/iphones. Anyone experienced this sort of thing? Any help would be greatly appreciated.

[Re-Titled by Moderator]

Original Title: Mail not sending