Is there a way to easily transfer email accounts from another hosting company as to not loose any data or passwords?

NO, there is no such a possibility. You'll have to recreate the accounts for sure, though you can set the same passwords for them as you had had before. You'll have to create all the contact lists again.

(a simple tip for transfer your server-side located emails, for not loose the incoming mails in the middle of the dns change process, I configure the two accounts trought IMAP in my email client software(set the servers with the IPs, to avoid dns change issues) and then I just drag&drop the emails from the old account to the new account. But of course you need to re-create your email accounts in the new server, as Lehrer say.)