The notion of a tie-in of military service with being a good citizen became fully developed in republican Rome. Back then, of course, Rome was either being constantly attacked by its neighbors, or later on in the republic, was conquering them. Naturally, having meaningful experience in the legions was seen as vital to a public officer or any significance; if you were intimately connected with warfare and all its contingencies, you could apply those to effectively handling one of the most important affairs of state - foreign policy, principally warfare.
This notion is also quite relevant to America. Since the inception of this nation to the early twentieth century, the primary way for an individual to ascend the governmental post ladder was through a distinguished military career, or at least substantial martial experience. This was not based totally on antiquated ideals. Rather, the military in those days (as today) instilled certain values which were (and are) considered necessary for a good public servant: honor, respect, and subordination. Transferrence from military connotation to civilian connotation is smooth. Honor for the soldier’s position within the armed services becomes honor for the politician’s office; respect translates similarly as honor; the soldier’s subordination to his superior’s decisions becomes the subordination of the politician to the dictates of those who had enlisted him, the populace at large (in the case of an appointee, such as a cabinet member, this is indirect but still relevant, or ought to be).
As to whether military service is still relevant today? I believe it is still highly relevant, but not a requisite for a good public servant. For those in war policy-planning positions, I would be more adament in wanting the individual to have had some experience in the armed forces. Although politicians can still be good leaders without such experience, they are more dispositioned to making good decisions (in wartime) if they have it. For example, take Dwight Eisenhower. He was the leader of the combined Allied forces, and did a superb job with that. He subsequently went on to be a strong, good president during one of the most harrowing times in American history, the Cold War. If it were not for his experience with WWII, he might have made a false step that could have led to an armed confrontation with the USSR - and let’s face it, those who have not actually been faced with war personally will more easily engage in it.
So, should it be required? No, not as a decree of the law. However, it ought to weigh heavily on the minds of voters, especially for those who would possibly be in a position to engage in a war.
And while this may seem tangential, I just have to address one of Blast Master’s comments:
Andrew Jackson did indeed have a highly-decorated military career, but he is certainly not considered one of our worst presidents. He’s actually considered one of the best ones, with an average scholarly ranking at the eighth best out of the 42 individuals to have held the office http://en.wikipedia.org/wiki/Historical_rankings_of_United_States_Presidents, because of his keeping the Union together during the crisis over the Tariff of Abominations, among other things. I just had to address that, because he’s one of my favourite presidents (right behind T.R., Wshington, Jefferson, and a select few others).