What about if you’re a public employer? I sense a school board fight coming, especially if the coach is a winner.
A good question, and this answer - whatever happens - is going to have multiple layers to it:
• This is a school in a rural area of Alabama ... next door to my own county, in fact. So I can say for certain the local community sentiment will be supportive to the coach... that area heavily (80-ish%) went for Romney.
• If he's a union member, the AEA (Alabama Education Association) is supposed to support him... but their knee-jerk reaction will be decidedly liberal, so they're likely to let him twist in the wind.
• The school system is really 'quasi-governmental' - operated by the government almost like a public utility. It tends to serve the public interests, so employees should probably hold any opinions to themselves to avoid hassles. But because this was done in rural Alabama, he might keep his job... after an apology and some 'training'... depending on how bent that school board is regarding 'diversity'.
• Overall, though, a "firing" is not the same as a "prosecution" for expressing an opinion, and thus I do not believe the 1st amendment applies here. Heck, administration officials are routinely fired (or they 'resign') after making comments embarrassing to the current administration.
Again: you're free to have an opinion... but you also need to weigh the potential consequences, for there are almost always consequences - and being right doesn't matter.