Your employer owns your work email account, you are merely allowed to use to it. If there are personal emails contained there, you should delete them. However, based on your company's backup policies that doesn't mean that they are actual no longer there.
Your company email belongs to the company, not you.
You should have no expectation of privacy regarding your work email account, voice messages, instant messaging, chat, etc. It may or not be legal for your employer to audit, review, listen to, or read your work communications, but you should assume that they will. If you don't want your employer to know your personal business then don't conduct personal business with work provided resources.
Never mix your work and personal lives, communication, etc.
Even in the USA, HIPAA protects health records from employers. A lot of folks here say don't mix personal with work, but in many cases I've seen, certain health benefits can only accept your work email for verification (ex you enroll in a health savings account and you get bonuses for completing health survey). Also payroll and HR information are usually to your work email, not personal. This is sensitive information not something anyone in the office can view which might contain your PII (SSN or whatever to your country), and pay info, etc stuff you don't share.
With that said, employers must protect these types of records. They cannot share it with everyone else as a record. Also, it would be crazy for you to inspect each email, and delete it before exporting it to a shared drive.
My advice: bring up the topic to your boss. Explain you have many personal health emails in it and check with your local laws to bring it to your boss's attention. Ex, "Under law X, an employer must protect my health files." etc so it's not just you saying it but something they must do. Ultimately, your boss may just want certain emails searchable so maybe you can just export those to the shared drive. I also recommend you remove association of any personal items like bank or social media from your work email before leaving. This means going into each site, and changing the email settings from your work to personal.
Legally: depends on where your employer is located.
From a technical standpoint: Yes. In fact, there's probably long term archives of all of your mail being created on a regular basis. These archives are just harder to get to than your boss wants it to be when getting the mail he might need.
I'd agree to help them because it's not an unreasonable request as far as company related messages go. Just go into your mailbox and start deleting any private messages you don't want your boss/co-workers reading. Even if your company keeps long term archives of mail somewhere, your boss & co-workers won't see them because they're only interested in easy to access mails placed in the shared location by you.
You can also ask your IT department what the policy is on any automatic archiving of email. At least where I've worked companies are usually very upfront about this on day 1 so it's not a surprise. Also, I live in the U.S. and there's always the potential of old company emails being accessed as part of a lawsuit.