As a first disclosure, I’m vegan. As second disclosure, I’ve personally faced situations similar to what you describe. Also, I assume that your bosses are closed to reason and slightly enjoy bothering you with that. I assume this because that is what I've found in my personal case. If my assumption is wrong, this may not be the answer you are looking for.
It seems almost unavoidable to find people with attitudes like that. However, there is a clear line between having a discussion (note, discussion is not a fight, but the exchange of arguments to find productive solutions and get closer to truth) and simply being mean. If what they do is to repeat hackneyed arguments, as you say, then to me it would signify that their purpose is closer to imposition and farther to discussion.
My suggestion is that when people do not want to listen (at least in this specific issue, they may be open to work along in other subjects) your energy is better spend educating yourself, challenging your own assumptions and making better arguments, not arguing back. Do it for you and for the people that do want to listen.
As someone here pointed out, if you already asked clearly to stop pressing the issue and they didn’t, then it’s harassment. If they don’t stop is because they don’t want it to stop. Asking differently may not be the solution. Running away from the problem only incentivize their behavior, something I don't think any vegetarian/vegan would like. If you don’t want to challenge them legally or aggresively and it keeps happening, then you have to get clever. You could mock about the irrationally of their arguments and how idiotic some of them may be, but this may escalate to aggression if done wrong.
What I’ve found (by experience) is that ignoring them altogether often gives them the courage to press further, until you break. What is the best answer for deaf ears then? Silence.
Not a sad/angry silence, just silence.
Not a look away/change the subject silence, but a ‘look directly and firmly into their eyes’ silence.
Active silence, confident silence.
Not a single word until the subject changes.
People is not used to this kind of reaction, nor to be looked in the eyes with impassive persistence. An additional slightly kind smile usually drives them crazy (not impassive, but also effective). They may think you are angry at first, but you are not. They won’t think you are afraid, because you are not.
I hope they change their minds eventually, or that you find a polite way to make them stop. But if they don’t change, and my suggestion does not work either, move on. Challenge legally or leave if you think it won’t work.