It seems hard to tell, given that private sector unions tend towards being at large quasi-governmental corporations. Unions do add difficulty to the firing process, which seems like it should be enough to fend off the employer firing you simply based on not wanting all of the harassing phone calls. But if management and union management still want to go through the process of firing you for what you said, then a government employee would still have the defense that doing so would violate their rights.
The prevalence of public sector unions itself is actually a good example of the trend I'm talking about though!
If you're such a screwup that even your union wants you gone, you'll lose your job, period. Whether it's public or private doesn't matter. Very often the job of a union rep or union legal counsel is to ensure a soft landing. Source: interned with a police union's legal counsel.
One thing to keep in mind about police officers and similarly situated government employees is that maintaining credibility as a court witness is a fundamental requirement of the position. A police officer who's credibility can be easily attacked on the stand has substantially diminished capacity to perform their duties. Part-and-parcel of being any kind of law enforcement officer is testifying to collected evidence on the witness stand.
While the government technically can't retaliate directly for not testifying against yourself in an administrative hearing, failure to be fully forthcoming in such an investigation demonstrates a lack of candor which in turn could be used to impeach you as a witness. If the government has evidence of lack of candor and sufficient motivation to get rid of you, you better start searching for a new job.
The prevalence of public sector unions itself is actually a good example of the trend I'm talking about though!