There have been repeated comments above to the effect that "I don't think a highly trained firefighter should have to pass out flyers at any public function."
This is an entirely separate issue from what's involved in the clipping and the founding post of this thread, entirely independent from any issue of prejudice, gender politics, or the like.
A fire department's governing officials are the ones who decide whether, for instance, the circulation of flyers at public gatherings serves the department's mission of minimizing fires and the dangers thereof. Clearly the decision of this fire department was this that this is an effective tool. Whether they are right or wrong in that is neither here nor there regarding the controversy involved here. They made that decision, and have the right and duty to use such publication as a public education tool.
Then too, a governmental unit always has to consider issues of public relations, and it will use (and rightly so) its employees in pursuance of that end, as long as the function doesn't get in the way of its prime function (here, fire safety). The publicly visible presence of department personnel, in uniform, serves both to lend authority to the educational matter in the flyers and to bolster the department's public relations.
Even if someone here disagrees with the official determinations and policies referred to in the previous two paragraphs, no-one not a consituent of that political subdivision has a right to question said policies.
I will not in this post take any position on the firefighters' motivations, nor on the higher officials' motivations in exercising discipline. I happen to think (based on what little I learned from the clipping) that the discipline imposed may have been excessive. But that's not my call to make, and it's not yours either, again unless you are a constituent of that political unit. Or unless you are a judge, properly presented such issues by way of litigation.