Send mail from OS X without email account
Yes, there's no problem in sending emails like that from OS X. 

First, you'll need to configure postfix to relay emails. Once you've done that, you'll need to create the script that sends the email. I'm not sure what method you're using to send email on the Linux system, but something very similar should work on OS X, for example, using mailx.

Assuming that your upstream mail provider (the system your OS X system is relaying mail through, ie the mail relay) doesn't require authentication, you won't need to configure mail authentication.