I would like to be able to assign the Zephyr test tasks that are linked to a User Story to a QA engineer when the story moves to the QA status. Is this possible?
Yep. Since tests are JIRA issues, they follow a workflow. In the transition to the QA status, add a post function to assign the task to the user in question.
Note that this will always assign the issue to the same user. If it's not 100% of the tasks that goes to the same QA engineer, you need another approach.
This is what I'd do:
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