Eich resigned because of external pressure on the Mozilla organization. I hear that one of the lobbying activities against him was when the dating site "OK Cupid" started informing Firefox users who accessed the site of Eich's activities and that they should download a browser made by people who don't nominate someone with gender discrimination issues to be their CEO. At the time, 8% of OK Cupid customers were there to arrange same-gender meetings.
They felt he was the public face of the company.
Russ Nelson published a piece on what he theorized was the economic motivation of Blacks to be lazy, and was booted off of the Open Source Initiative board. He wasn't thinking about how it would be perceived. A modified version of the piece is still online, but not the version that got him in trouble. In general, executives are seen as the public faces of their organizations even in the case of Nelson, who was not the chairman of the board, but was simply a member of the executive board. In Nelson's case, it wasn't that he made publicity appearances and press releases, it was that he was one of the people with the power to direct the company (and thus a more real face of the company than soneone who just does PR), and folks did not trust that someone who wrote what he did would behave as they would like in that position.