I've experienced from time to time a problem with webmail functioning after cPanel updates, particularly on larger servers. You can reinstall all three webmail applications, while not disrupting user content, with the following commands:

Roundcube:

/usr/local/cp

The WordPress Classroom

anel/bin/update-roundcube –force

Horde:

/usr/local/cpanel/bin/update-horde –force

or

/scripts/fullhordereset

Squirrelmail

/usr/local/cpanel/bin/update-squirrelmail –force