Does the Cypress-Jira Integration work with private Cypress projects?

No, at the moment, the integration only works with public Cypress projects.

I'm asked to log in, and then I'm getting an error " refused to connect"/"Firefox Can't Open This Page". What should I do?

This is a sign that your Cypress project is private. Please log in to your Cypress account in a separate window and set the project to public under "Project Settings" - "Access".