I have a hard time seeing how anyone can deny that drafted soldiers are slaves of the state. They are giving their time and labor only under compulsion, and while they may be better off than ante-bellum slaves in that they may eventually get freed after their term is over, to some extent they may be worse off as their time in servitude is a) more dangerous and b) involves taking morally more questionable actions (e.g. killing people).
I have assumed that those who supported the draft either were arguing that that threats in wartime justified this awful step or they were statists that already saw all the rest of us as slaves anyway.
However, Bryan Caplan had a useful observation on this:
It's tempting to dismiss all this as doublethink, but after many years of reflection I think I finally figured out what most people are thinking. Namely: They implicitly regard slavery not as mere involuntary servitude, but as low-status involuntary servitude. Since most of us honor, respect, and even adore all our soldiers, conscripts have high status - and therefore can't be slaves. From this point of view, saying "conscription is slavery" isn't righteously standing up for the rights of conscripts; it's wickedly denying them their high status. Sigh.
This rings true to me, but offers another avenue for those of us who oppose the draft -- the draft reduces the perceived status of those who serve voluntarily, something I certainly think happened in the Vietnam War. In a way, it is reminiscent of how the existence of affirmative action tends to undermine the perceived accomplishments of successful minorities.