Absolutely no right means "lack of any right whatever, lack of any right at all". Absoutely modifies no and characterizes the lack of right as absolute—the lack is unqualified under any circumstance.
No absolute right means "lack of an absolute right". Absolute modifies right and characterizes the right as absolute—the right is unqualified under any circumstance. No absolute right implies that some sort of right exists, but any right that exists may be qualified under some circumstances.
From a purely linguistic point of view, the State's argument is without merit. (This does not, however, preclude the existence of statute or case law which overrides the linguistic point.)