A new version of the Shotgun-Jira Bridge is out. It is a small bugfix release. We updated the Jira authentication to use basic_auth instead of cookie-based auth. Jira Cloud requires users generate an API token for authenticating, as user passwords are not usable with basic_auth on Jira Cloud. See Manage API tokens for your Atlassian account | Atlassian Support for information on how to generate a token.
Have a nice day!