What is the geologic origin of the Bauxite deposit in Bauxite, Arkansas, and could there be more?
The short answer is, bauxite requires a particular alumina-rich source rock and a specific set of conditions and processes to concentrate the aluminum in a specific order.

From the  Encyclopedia of Arkansas:


  The Arkansas bauxite region covers about 275 square miles in the northern part of the West Gulf Coastal Plain and is divided into two mining districts. One area is in Pulaski County, south and east of Little Rock, and the other is in nearby Saline County, northeast and east of Benton.


 

(Source, Arkansas Geological Survey; a very detailed map can be downloaded here)

The source rock (nepheline syenite) for the bauxite appears as two isolated blue spots in the middle of the map above, surrounded by (orange) Tertiary sedimentary deposits. The town of Bauxite is just to the northwest of the southern spot.

Rather than cut and paste EOA's description of the process, I decided on the following summary:


Magma was intruded during a Cretaceous volcanic event as the Mississipi Rift opened, and this formed nepheline syenite, which has aluminum-rich nepheline where granite would have quartz.
The syenite decomposed, eroded, and was deposited as a soil on the edge of a shallow sea that filled the rift.
In the Paleocene, tropical conditions caused the aluminum minerals toleach down into the soil, becoming concentrated into a hard layer called a laterite
In the Eocene, some of the laterite was eroded and deposited as alluvial bauxite ore as the sea slowly filled in.


Both the original laterite and the alluvial deposits have been mined for aluminum.

Your assertion that there are no other bauxite deposits in the US is not true; other deposits formed in similar conditions in the southeastern US, and there is also bauxite in Montana.