The southeast equivalent of White Castle is Krystal burgers. They were in Texas for a while, where I experienced them, but no longer there.
Same "buy them by the sackful" concept.

https://en.wikipedia.org/wiki/Krystal_(restaurant)

California does have White Castle burgers ... though just in the frozen foods section of my local grocery store. Frozen / thawed / re-heated White Castle burgers are NOT for the gourmet palate. Personally, I wouldn't feed the frozen ones to my dog. The fresh-made ones, like at Casino Royale in LV, are edible, but an acquired taste. Maybe one time, for the bucket list, then move on.