It's an SSL issue that takes a few steps to fix.
1) Go to the ISP of each email account and find the actual server name for your account. Change your incoming and outgoing email servers from mailorpop.mydomain.com to mailorpop.servername.com. The server name often has a number in it. (e.g. mail.xserver356.com) Do this for each account.
2) Quit Mac Mail. Open Keychain Access (Applications --> Utilities). At the left click on Logins, then below click on Certificates. Find all email certificates, hold the option key down while clicking on them, then select Delete to get rid of them. Quit Keychain Access.
3) Open Mac Mail. For each account, you will be asked to enter your password again. Click on the Certificate button at the bottom of the window, click the Trust triangle, then check Always Trust. Do this for all incoming email. Then send a test email from each account, and do this process for all outgoing email.
4) Open Keychain Access and find your new login certificates. Double click each one, then click the Trust triangle, and select Always Trust on the top scrollbar. Close the window and enter your keychain password to Update. Do this for each email certificate.
It's a pain in the neck, but after all this, your email should work fine with again with SSL.