They have an obligation to operate according to the precepts of the Church; from this, and only this, does a government have the right (not the ability) to rule and do so justly.
Would you agree that...
If they do not have a RIGHT to rule, I do NOT have a duty to obey?
That would depend on what you're obeying. At least in the abstract, I would argue that the faithful would not be bound to obey that government. In practice, that would depend of the situation. I would argue that it would be licit to follow those laws that are just, or that are in agreement with Catholic morality (prohibition against murder), and to disobey those laws that would be harmful to faith or that make us a party to sin (such as paying taxes to a statist government that use it to provide euthanasia, and chemical or surgical abortion). Moreover, Thomas Aquinas asserted that it is sometimes necessary to tolerate particular evils to prevent chaos, as it is for faithful Catholics who must live and operate in secular societies that grow more hostile toward the faith with each passing year. Acceptance of an unjust power's temporal authority for the sake of order, to suffer this while speaking against it peaceably is sometimes necessary; such is an example of the virtue of fortitude.
To put it simply, if obeying an unjust or unlawful secular power does not require one to sin, or to be a party to sin, and to do so will maintain or improve the common good, then it can be considered just to do so. If rebellion against that power can be had with reasonable assurance of both success and that the common good would not be made worse by it, then I would argue that rebellion is just. If reasonable success of revolt is impossible, then to refuse to sin by following its precepts, suffering what punishment such refusal would engender, and to preach the truth of Christ against that unlawful power seems the most wise and just course.