Yes. The client user will be able to change the issue status only if an issue is assigned to them.

To assign an issue to the client user, the following two conditions should be accomplished:

1. The Assign external issues to client users option must be enabled: Once you are in the Issues module, click Issue Settings --> Configuration menu and enable this option.

2. The issue's flag must be external: You will be able to see the list of client users while filing the issue only when the External flag option is chosen.  Learn more.