Welcome to EH and the forums.

Pretty sure no one here will help you to hack into your gmail account as a proof of concept if thats what you are asking for.

Forwarding internal company data (including patient details) onto gmail or any other similar webserver is no good idea at all in my opinion.

There are quite a few risks associated with this. Just consider all the possible attack vectors such as sniffing the forwarded emails etc.
Also as gmail is a webbased service you can log in from everywhere. So when someone logs in there is the possibility that the machine which is used (e.g. at home) is infected with a malware such as keyloggers.

Once an attacker got access it is easy to change all passwords etc. and you won't be able to login anymore while the attacker gets access to all emails stored so far and depending on how fast it is detected (e.g. when the password is not changed) to all future ones.

There are many more risks associated with this approach, at least in my opinion.

Thinking as a patient I would not feel comfortable with this too.

Like Don said. We all feel the pain on these occasions, but trying to own Google Mail isnt going to do you or your company any good.

Compliance plays its little part here, so look to your external regulators and internal policies to argue the case. If you dont have this, your probably in for a hard time anyway.