I dont have a solution, but wonder how you see this working;
The content of the mail is what, static text? the user just replies inline or at the top of the message (emails are not structured), makes analysis of the response hard to aggregate, but OK if its 'how was your issue' type feedback.
The responses go to a different user, back to JIRA for comment (you probably wouldn't want the issue reopened, or have the remote user notified?)
While it is certainly possible to send an custom email for issues that fullfill a certain JQL query (status = closed and updatedate = -7d) it's how you are going to collect the replies that will be the challenge (like Andy rightfully indicates).
You could do this using a custom transition (from Closed to Closed) that triggers an event that will send out a custom email. Auto Executing this transition could be done using jelly scripts and a jelly service for example.
You could ofcourse send the customer a link to an online form where he could fill in the questions (you might even be able to populate certain hidden fields in the form using url parameters (issue id for example).
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