Well,
Both OpenWebmail and Squirrelmail use IMAP to connect to mailboxes (although depending on the exact install, OpenWebmail also tries to write stuff directly in the user's homedir).
I'm assuming you want to move the entire e-mail functionality to the new server? As long as you manage to move all the imap folders and data from the old mailserver to the new one, that should work just fine with no loss of functionality. (though I don't think it's a trivial operation; I've never tried moving a whole bunch of e-mail users and all their data from one mailserver to another, but I can't imagine it'll be easy)
Address books and customized junk is going to be a different story. If the users want to use their addresses, they'd better export them from Openwebmail and then add them by hand in Squirrelmail. AFAIK there's no automated way to do that.