I have not delved deeply, but it seems that Jira has an elaborate REST API (https://developer.atlassian.com/jiradev/jira-apis)
So the best way would be to find if the Jira Stories can be fetched with a REST call.
As Ronal suggest use the API of Jira to extract userstories from there and use Mendix Stories API to create/update them in the sprintr environment of Mendix by using the following documentation:
Both answers suggest using the REST API that comes with JiRa and allows you to manipulate stories in the tool.
Although this is the most flexible solution as you can customize it as you see fit it is not the simplest one.
Jira supports something called issue collector. It is very similar to the provide feedback button that you get with the feedback widget.
The good thing is that al you need to make this work is :
1. to register your JIRA project with the issue collector in jira,
2. set up your issue template (or go with the default one), and choose the look of the feedback button and popup window
Done. It takes less than 15 minutes and works like a charm.