Well, you can fire a union employee, but it takes awhile. Verbal warning, written warning, work plan for at least 90 days, monthly (or oftener) evaluations; then decrease in pay, demotion or termination. A former supervisor I know says "If you do it right, you can fire God!"
The work plan has to be measurable, ie failure to perform duties, bad language, not getting along with others, complaints from clients or patients, etc; it can't be for a "bad attitude" or the like.
Gross insubordination, fighting, physical abuse can speed up the process, if you have proof. Of course the individual receives a copy of the work plan and of each of the evaluations.
We went through this with a former HN who did not follow all the steps, the fired employee sued and won enough to be able to return to school for a masters'. He's now a nurse entrepreneur. (I never thought she had grounds anyway!)