Either one is fine, but "home" might be slightly more appropriate in that it could be interpreted to include apartments as well as houses.

The word house is used to refer to a building made for a family or people to live in, whereas a home refers to the family living in a house or the place someone was born or lived in as a child. So, if the speaker is referring to buildings that people live in in general then the second option "a house" would be best suiting

Both are correct, but A is better. It would be still better to say "someone's home" because the reason for taking off their shoes is not that they do not like to wear shoes in homes or houses. The reason is to show respect for the person who lives there. Saying "someone's" emphasizes respect for that person. The reason that "home" is a better word to use than "house" is that a house is merely a building that is located in some particular place. "Home", however, is a place in the heart.

They always take their shoes off when they enter someone’s home. (‘home’ is the better word, but the sentence as written is unnatural)