Can Tasmanian Crayfish grow in the US?

No. All species of crayfish which are not native to North America (European, Asian and Australian too) are susceptible to crayfish plague. The disease is caused by a water mould present in all America. Crustaceans die within few weeks of being infected.

What's more American crays are not only immune to the plague but also are carriers of the mould. This fact causes severe problems in Europe, where some American species were introduced (signal crayfish, swamp crayfish, belted crayfish) and now are increasing in number and range bringing European crays (bigger and of higher economical value) closer to the brink of extinction.