If a new CEO of Coca-Cola made a speech announcing a proud partnership with the KKK or Al Qaeda and ran an ad campaign informing the public then sure I imagine that'd be a lawsuit. Coca Cola would never disassociate its brand, every reasonable person could predict the forthcoming disaster and every shareholder would want to recoup their losses.
So yes, CEOs do have a duty to avoid saying certain things, sometimes. Does it extend to Trump? Not quite, I don't think, but the very idea of it is not unreasonable.
Last time I looked, Trump's support and Clinton's support were pretty close to evenly split (Clinton slightly ahead in most polls, but not by that much).
Why is supporting Trump more "tarnishing" than supporting Clinton?
I understand that you don't like Trump (which is fine, I don't much like him either), but why do your opinions get to control? Do you think there aren't people who find Clinton equally repugnant, or even more so? I assure you that there are (I'm not voting for either, myself).
But, you know, the money of Trump supporters is just as green as the money of Clinton supporters, if you want to cast it solely in terms of fiduciary duty. Probably even more so, since I would expect that the median Trump supporter is wealthier than the median Clinton supporter.
I don't know, I think most of the money is on the left side of the aisle this cycle. Clinton has firmly grasped the title of "establishment" ie status quo, and status quo is good for the wealthy. Many, many wealthy republicans have flipped, IIRC from polls I think Trump's poor white base would dominate the median income.
So yes, CEOs do have a duty to avoid saying certain things, sometimes. Does it extend to Trump? Not quite, I don't think, but the very idea of it is not unreasonable.