Dose my employer have the right to make me write down my problems with other employees . Five goals for improving my behavior even though I have never admitted any wrong doing.
Employers can do pretty much whatever they want, unless it violates some law. Was there a reason why your employer asked you to write down these goals? Did you have a problem with co-workers? Did you discuss with your boss why you got this assignment? Writing down goals for improving behavior doesn't necessarily mean you did something wrong, but it might mean that your boss wants you to improve. Based only on the information you gave, I advise you to start writing -- or look for another job.
This does not by itself seem to violate any laws. But, if it is targeting employees based on a protected class or activity, there may be some illegality