There are several issues here to be addressed. Yes indeed your boss has the absolute right and responsibility to tell you what is is not appropriate in official business correspondence. Anything coming from the company, like responses to issues that customers have brought to a help desk, needs to enhance the company image in the way the company wants it enhanced. Even internally, what you write in official systems is part of the impression that other parts of the company have of your department not just you as an individual.
And it is entirely possible that your boss was asked to get you to stop adding those phrases because there were complaints about them. Many people absolutely hate the phrase "Have a nice day." It is not being nice to add that phrase. At best, it is a meaningless stock phrase. At worse it is actively at odds with the message being answered.
It is a good thing to want to be nice to the customers or clients. What you need to address is a more effective way to do that than the use of stock phrases. If you are in a help desk situation (which it sounded like you were), then the best way to be nice to the customer is to fix their problem. The next best thing is to actively listen to what they have to say and respond to it appropriately without getting mad or snippy or going off on an unrelated tangent.
For instance, I recently had a question about something for a piece of software I used. The help desk person responded with, that information is on our website. Well if I had been able to find it on the website I would not have asked. I pointed this out and the response was a link to a 60 page pdf document. I pointed out that I didn't want to wade through that and could he just answer my very simple question (BTW I did read it and it skated around my issue but did not answer it.). He pointed me to something related to my question but not answering it. Something far enough away from what I asked that if he had paid attention to what I asked he would not have sent that to me as an answer. I then emailed back and told him that I would no longer be buying further software from his company.
In any of those emails, "Have a nice day." would have been taken as sarcastic nonsense. My question was one he should have been able to answer in five minutes or less. Instead we went about two weeks emailing back and forth as I tried to get him to actually pay attention to what I was saying. A customer service rep who was being nice would have simply answered the question in the first place. One who didn't know the answer, should have referred me to someone who would or looked for the answer himself. That is being nice, not using stock phrases.
Now another way to be nice is to respond to what the person says when they go off on a tangent. For instance, I totally lost it once when a company kept calling my phone and asking for my boyfriend who had just died. The third time I told them he was dead, I totally lost it and begged them to take him out of their database. A nice customer service rep at that point (and I did have one, it wasn't his fault that no one had marked my boyfriend as dead in their system) would have expressed sympathy for my loss as well as taking action to ensure I didn't have to endure frequent calls from this company asking for a person who was no longer alive. A not nice customer service rep would have ignored what I said. Sending me an email confirming that they had removed my dead boyfriend from their marketing database that ended with "Have a nice day." would have sent me into tears. It would not have been a nice gesture.
With customers such as internal ones that you deal with repeatedly, you can get more personal, but probably not within an official ticketing system. If someone just returned from maternity leave, I might ask about their baby. Someone who just got back from a vacation in Hawaii might get asked how the trip went etc.
Connecting on a personal basis is being nice. Going the extra mile to help someone out is nice. Smiling at people and complimenting them is nice. Telling their boss what a a good job they are doing is nice. Calming down someone who is upset is nice. Using rote phrases is neutral at best.
We won't be solving your problem for various reasons. Have a nice day!)