Lots of problems here. In no particular order.
a) Modern warfare is all about coordination and training. Look at the number of friendly fire incidents we've seen with just regular forces. Having new and uncoordinated troops muddling up the battiefield would be a problem.
b) A mercenary company would not necessarily be subject to, or adhere, to the US military code of conduct. We’re trying to give the impression that we’re the good guys. So having a troop of hired yahoos drive into a village, looting and molesting the women would be a very bad thing.
c) Historically, after the war is over, you have the problem of the mercenaries hanging around with time on their hands, not much money, and lots of weapons. That’s a bad combination.
d) We have something like 90,000 troops on the ground in Iraq with maybe 250,000 support personnel. (Numbers very approximate.) So what’s the ad in Soldier of Fortune magazine gonna look like? “Needed, army of 350,000 to invade small but heavily armed country. Bring your own armored vehicles and RPGs.” There really isn’t anyone out there in the business on that scale.
e) Problems with command and control. Lots of mercenary companies (at least in the Middle Ages) had their own charismatic commanders. There was lots of bickering and rivalry among the commanders. So getting them to behave and do what they’re told might be a problem as egos collide.
f) Ask yourself what kind of person is going to sign up as a mercenary? In the armed services of a country, there are lots of reasons (besides conscription) to sign up: training, patriotism, financial incentives, and it’s a reasonable career path with a fair chance that you’ll never actually get into a shooting war. In a mercenary company, you will get shot at and there’s no job security when the job is done. So you’ll get the desperate, the psychopathic, the suicidal, and maybe a few idealists. (See section b)
g) Loyalty and motivation. A mercenary company is loyal just as long as the checks keep rolling in, and as long as no one outbids you. And, when things get tough, it may occur to the mercenary that his or her skin is worth more than that paycheck. So you can expect more desertions (See section b) and risk having a large chunk of your fighting force suddenly decide that they’ve gotten a better offer.