It IS a custom in other parts of the US for folks to remove their shoes when entering a home. That's apparently not the case in Texas, which is fine, but it's hardly rude to follow the local customs--whether it is an Asian custom or another custom of a different region of the U.S.
I did a quick search and found this about why the Japanese remove their shoes:
Why Remove Shoes ?
The custom of removing outside footwear within the house goes back at least as far as the Heian period ( 794 - 1192 ) among the upper classes and gradually spread thereafter throughout society. One of the reasons that footwear was shed in this fashion was because of the high rainfall and the generally very damp climate. A house would be quickly dirtied if people walked in wearing mud-covered shoes or sandals. But probably what came first was the custom of both sitting and sleeping directly on the floor on straw mats or cushions laid over it. Footwear was removed at the entrance to help keep the house clean. http://www.tjf.or.jp/eng/content/japaneseculture/02kutsu.htm
The reason the Japanese do it for cleanliness is because, at least until the 1980s, they barely used furniture. They had/have expensive, handwoven mats lining their homes (tatami mats) that they sat on, ate on, and basically lived on. They live on their floors like we live on our couches.
I agree that the reason for shoe removal for "everyone" is for cleanliness reasons, but the Japanese take it a stricter form because they are on their floors all day.
My Asian friends gave this explanation as well. In many Asian homes, furniture was/is low to the ground and quite minimal, i.e. mats, pillows, etc. that people both walk and sit on. I feel that it is an acceptable request to ask someone not to walk on the articles of furniture that also serve as chairs.
Again, IMO, floors are made to be walked on and will get dirty. I just don't think it's right to put my floors above my guest's comfort.