Speaking as someone who does eat meat, there are a few key points you might want to consider.
Vegans and non-vegans are not a different species
Rather than viewing the situation as a "them and us" scenario, it's worth remembering we're all people. The viewpoints might differ but we all ultimately work the same way.
No-one likes "holier than thou"
No matter how superior you may feel (if at all - you may well not) to someone who doesn't follow a vegan lifestyle, you won't change any minds by putting yourself forward. Instead, you'll either come across as arrogant (in which case why should anyone listen to you) or be perceived as setting impossibly high standards (in which case, what's the point of trying).
Challenge obliquely
Confrontation won't work since ultimately the person with the power over whether an omnivore changes their habits is the omnivore themselves. Since you don't have the authority to dictate, don't try to do so. Instead, lay out reasons for considering a change of attitude and let them mull it over. If they don't change, they weren't going to by threats, insults or dictats either and so you've lost nothing. If they do change, you've achieved something. "Have you considered..." and "But what about...." are your friends.
Incremental change is better than no change
Focusing on areas where, if a person were to choose to try and change, such change is relatively easy means you may well be able to encourage change by degrees. If someone decides to eat meat less often that is still less meat being consumed. If someone decides to buy meat more ethically, that is still a reduction in cruelty. By no means ideal by your vegan standards but an improvement on the current situation. And once that change has been incorporated into an omnivore's life, their base point has changed meaning focusing on the next easiest change to make becomes less of a leap for them.
If you've got to talk ethics, talk about yours not theirs
I have rarely met someone who likes being preached to or sold to, so telling someone where they are wrong is never likely to end well. However, taking a stance like "I found myself becoming increasingly uncomfortable with....so I started doing things this way instead..." simply means you're commenting on yourself. It says what you found difficult to live with and how you resolved that conflict. It's fine for you to judge yourself and by not judging anyone else by your yardstick means they are free to draw their own conclusions. Bear in mind you have a viewpoint that makes sense to you but that doesn't mean that viewpoint is "right"; right and wrong are remarkably difficult to define in absolute terms, so best not to try. "Works well for me" is much easier to assert.
Different people, different priorities
We've seen environmental campaigners attacking owners of diesel vehicles without bothering to find out what else hidden to them someone may be doing to improve their environmental impact. I had a choice of installing solar panels and an air source heat pump or buying an electric car; I couldn't afford both. All anyone sees however whilst I'm out and about is the diesel car, yet I have reduced my household's CO2 emissions by more than the car generates. Same goes for changing people's minds about diet; take time to know a little more about what positive steps they are already taking and celebrate what they are already doing right before tackling what you see as things that still need to change.
Demonstrate your willingness to be open-minded
If you aren't prepared to show willingness to change in the face of a well-reasoned argument, why should anyone else in the face of one of yours? That doesn't mean you have to compromise on your principles but simply saying something like "That's an interesting point, let me think about it" and then following up later with your REASONED response demonstrates admirably that you're prepared to listen. That will influence much more than simply expecting others to listen to you.
Finally, be consistent and accurate
If your views are built upon dodgy logic or if you are sloppy in the facts you provide, your arguments will fall down and you will develop a reputation as someone not worth being listened to. No matter how much you like that Youtube video or that blog article, if you don't research its sources to ensure it's not just another conspiracy theory then you'll end up having the rug pulled out from under your feet by someone who WAS prepared to put in the effort to cross-check. Eventually, lots of people saying the same thing doesn't make that thing right; it may just be something wrong being given a lot of air time. If you can back up all your assertions with reasonable factual evidence, the points you make are immeasurably stronger.
Overall, what I'm saying is in fact nothing to do with vegans talking to non vegans per se. It's really just a 101 on how to interact with other people and put your point across, but it's none the less valid for all that.
,
