Short answer: Yes it will cause issues for the Zephyr Cloud add-on. Maybe you can restrict that users access to projects you don't want Zephyr to touch instead?
Longer (more technical) answer: Atlassian Connect is the framework that all ecosystem developers (like Zephyr) must use in order to write plugins for Cloud. In order to give these add-ons the ability to perform actions in Cloud the Atlassian Connect framework creates users on their behalf through which they can perform actions. When the add-on's request permissions (at the time that they are installed) and you agree to them then the service user is created for the add-on.
If you were to create the user then the Connect add-on would no longer have the scopes required to perform any actions or queries against Cloud. So I don't recommend doing that; you'll probably have to reinstall the add-on to fix it.
Edit: The good news is that the team in charge of User Management seems to have thought of this already:
Screen Shot 2015-04-13 at 8.55.24 pm.png
As you can see you cannot deactivate or delete the user. Though you can modify their groups, and thus the permissions, of the SYSTEM user.
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