I'll argue that, no, they shouldn't. It is my viewpoint that there isn't any objective moral "truth". And that depending on your goal or values, your morals will be different. I was under the impression that this what ethics professors teach: all the different viewpoints, how to arrive at them, and how to argue for them.
This would necessarily put ethics professors into several different ethics standards themselves. Some of which would be impossible to hold and follow.
And even if you do deem something as "moral" that doesn't necessarily mean you can go about doing it. Taking the eating meat example: What do you expect the professor to do in a society where meat is the only available food source? Starve? It's natural to do certain things (as much as I hate the naturalistic fallacy). And whether or not these things line up with some definition that we call "moral" is irrelevant.
There's also the problems where it's difficult or impossible to figure out what the "right" thing to do is depending on your morals. The fat guy vs many problem is one that comes to mind.