Does the government have the right to institute a draft in times of war? Include 1 core democratic value.

Firstly, governments do not have rights.  Rights are the purview of individuals.  In the US, at least until recently, government derived its power by the consent of the governed, but this should not be understood that individuals ever ceded rights to the government -- rights, by definition, are intrinsic to an individual by virtue of existence, and cannot be altered or taken away by government. Secondly, the proper role of government is to safeguard individual rights.  Your question may be more properly stated as "Does the government have the power to institute a draft?"  Certainly in the US it has done so in times past, beginning with the North - South conflict in the 1860's. Establishment of a draft then resulted in riots in NYC that left many dead.  Although acts have been passed to institute a draft, its Constitutionality has been questioned several times, because it allows the power of the state to coerce an individual into the military.  Amendment 13 forbids involuntary servitude, and was made part of the Constitution as the final remedy to any kind of slavery in the US.  During the Vietnam Conflict, this amendment had been used once again to argue that the draft is a form of involuntary servitude, and apparently draftees who argued that their sovereign individual rights had been violated by being drafted were either released from or  were not forced to join the armed forces.