The metaphorical "lizard hindbrain" of common law, the basis of US law, has a lot of powers the governments can enact in cases of public health, both de facto and de jure. While they are not allowed to extend it past that point, they have a lot of existing power in this space.
And... honestly... not a lot of sensible people are going to complain. Only the very fringes are going to object. The vast "middle" majority, in this case 98%+, is going to agree, conform, and be upset at the people objecting to the quarantine and who break it, not at the government. That counts for a lot too, in practice.
This site leans incredibly libertarian and most people are not going to have a problem with this at all. I've seen people objecting to CDC existing on the basis of federalism and the right of assembly and all kinds of other stuff and that's just so far off the reservation of well-decided legal doctrine that it might as well be freemen on the land.
It's frankly a very good idea. I'd even say to lower the number to 50 or maybe 100 at most. Some of the gatherings in South Korea that are believed to have been super-spreading events have been less than that (that church group).
And... honestly... not a lot of sensible people are going to complain. Only the very fringes are going to object. The vast "middle" majority, in this case 98%+, is going to agree, conform, and be upset at the people objecting to the quarantine and who break it, not at the government. That counts for a lot too, in practice.