Well, I'm all for the integration of women into the military, including combat roles. For many of the requirements of modern combat, a women can do the job as well a man - especially a modern women raised in sports, to control her emotions, and to value herself as the equal of a man (in other words raised equally with men). The modern military runs on effective error free communication, attention to detail, rapid and judious decision making, and good planning - all of which I have no reason to believe that women can't do as well as men. I have every reason to believe that our military, which I have the greatest respect for, increasingly believes the same and at the very least teaches this to its recruits. However, there seems to be some resentment despite this, which I find completely impossible to understand.
"We're not allowed in the special forces either like Navy Seals. So for country that emphasizes equality, there's not much for women in the military.
I don't know about you, but it doesn't matter what gender you are, the gun doesn't care who fires it. Also if you know martial arts very well, its not strength that makes the best fighter, it is speed, agility, and intelligence to know where to hit for maximum damage with little effort.
There was a time when women fought with men in battle or commanded men in battle (see Vietnam 10AD) but then religions that did away with the female god to fill it with a male god kind of killed it."
Now, I don't think that there is a significant difference in the accuracy that is attainable by a female with firearm and a man. Witness famous sharpshooters like Annie Oakley. Witness my grandma before her health failed, for that matter. But there is a whole lot more to being combat infantry than the ability to pull a trigger and hit what you are aiming at. No matter what age you are talking about, full combat gear for combat infantry runs between 60 and 100lbs, and although this isn't a very popular thing to say there aren't a whole lot of women that can handle 80lbs of gear as comfortably as a man. I'm sure there are some, but the small number that can does not justify the extra complexity of a mixed sex infantry platoon/squad whatever. And, again, Hollywood movies not withstanding, there are NO women on the face of the earth that can qualify as a SEAL. There are very very few men that can. The physical requirements are just way too high.
Furthermore, any person that tells that speed, agility, and intelligence can make up for strength in hand to hand combat hasn't done alot of hand to hand combat - or for that matter doesn't understand how closely strength and speed are related. I'm not the greatest fencer in the world, but I can hold my own against the a collegiate female fencer that is accounted quite good though I'm meat for her male counterparts. And in physical combat, the advantage of strength and mass would be that much greater. Yes, there are some female boxers that would probably lay me out on my back, but there are no female boxers than can compete with the best male professionals, just as the Serena sisters (good as they are) would be decimated by even a relative low ranked male professional, or a decent men's senior player. Not that hand to hand combat is that important in modern war, and if it were only that it probably wouldn't matter.
Given all that, I don't think there has ever been widespread and significant female combat infantry anywhere on the planet, nor do I see a reason to begin now. But, in today's army, you can fly a fighter, serve aboard a warship, or pilot an armed remote drone (and probably any number of other jobs I'm not aware of) if you really want to be directly involved in combat; and, no one can say that the job you did was less important than any other job in the armed forces.
As for the original question, you can serve in the US Armed Forces if you are not a citizen provided you are a legal permenent resident (and indeed it speeds the process of gaining citizenship), though I have no idea why you would want to unless you wish to become a citizen of the USA. My general responce to this is, would you serve faithfully if the US invaded Finland? Not that there is any danger of us invading Finland, but if in your heart the answer is no, then please don't join. The oath is that sacred. If you aren't willing to give up your other loyalties, don't take an oath that you might be tempted to break at some point.
On a final note, don't join the French Foreign Legion. It is not nearly so romantic as it is made out to be. The equipment is tradiationally lousy, the assignments horrible, the indoctrination brutal, and the death rate in combat historically frighteningly high.