Yes, you can definitely do this, but there are a few things to be done.
Under Email - Email Configuration - Receiving Email - Anti-Relay Settings you will need to add the new domains you want the appliance to accept mail for
Under Email - Email Configuration - Sending Email - Domain Routing you will need to add the Exchange servers that the appliance will relay mail to for these domains.
These hospitals we acquired have their own MX Records and not part of our domain and we can still have them deliver email to our Inbound appliance without them being a part of our MX record and by just adding their domain or MX record? How would this work with quarantined emails on this box? Would we just add the domain and proper authentication for that domain to our appliance?
You will have to modify their MX to point to your MEG, but other than that, the MEG can accept and process mail for unrelated domains. You will add the domains following the above steps. On Box quarantine will work the same for these domains as it does for yours. If they have their own LDAP/AD setup, then you will need to add those servers in to the LDAP configuration for the appliance so it can talk to them.