As advised below part of JSP-38018, would like to get request for a feature, please
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David Yu [Atlassian] updated JSP-38018:
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Resolution: Answered
Status: Resolution Suggested (was: Atlassian Investigating)

Hi Lokesh,

After investigating this for you, there isn't a way to remove sharing with Everyone unless you modify the source of JIRA.

If you would like to see this feature implemented in a future version, please submit a feature request at:

http://jira.atlassian.com

For more information on how Atlassian implements new features and improvements:

http://confluence.atlassian.com/x/AfAB

Best Regards,
David Yu
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Please let me know if need any more details

Thanks