First off, there are no hypersonic missiles and will not be for a good 5 or 6 years at least.
Secondly, swarming with missiles is, indeed, one way to kill warships. Its exactly what the Soviets planned to do in order to prevent the US from reinforcing Europe in the event of a NATO/Warsaw Pact war before the end of the Cold War. However, the US is very, very, very good at fighting this sort of war. A Zumwalt has 80 VLS cells that can be packed with missiles (though these are meant for attacking, not defending on the Zumies, except when using the ESSM self defense missile (4 to a VLS cell)).
However, the Burke class, which would accompany a Zum, each have 90+ VLS cells themselves and an excellent radar system. You'll probably need a minimum of 50 to 75 missiles to get a ship for each Burke defending the Zum. It starts getting really expensive. The US is really good at the hot missile on missile action: even against ballistic missiles.
The best way to attack a Zum is with a sub. Defenses against subs are not nearly as good as against incoming missiles. There are NUMEROUS examples of even 'bitty' SSK subs sneaking up on even the big carriers.