In The Mixed-Up Files of Mrs. Basil E. Frankweiler, why do you think Claudia would not admit that she had said the words, "hide out in"?

I think that Claudia would not admit that she had said the words "hide out in" because she did not want to admit to being wrong, but was too focused on the matter at hand to really think about what she had just said.  If she had taken the time to think, she would most likely have realized that Jamie's accusation about her grammar was incorrect, but in the situation, she really did not care enough about it to come up with a good counterargument in her own defense.

It was Claudia who actually started the heated exchange.  An A-student who tends to be a bit of a know-it-all, she has the annoying habit of correcting the grammatical errors of others at every opportunity.  In the conversation in question, Claudia had asked Jamie, "What woods?", to which he had replied, "The woods we'll be hiding out in".  Claudia had immediately jumped on him for using a sentence with a preposition, in, at the end, which is grammatically incorrect.  She then asked Jamie, "Who ever told you that we were going to hide out in the woods?".  Understandably angry, but not as versed in grammar to be able to get his sister back effectively, Jamie accused Claudia of saying the same thing for which she had criticized him, the word sequence "hid(ing) out in".  In actuality, Claudia's use of the phrase was correct, because she had not, like Jamie, ended the sentence with a preposition.  Claudia was too focused on the task at hand, however, to really think about that. "Trying hard to remain calm" as befitted a leader of an escapade such as the one they were trying to accomplish, she tried to get out of the argument by simply refusing to admit being guilty of incorrect grammar.  When that didn't work, she managed just to change the subject (Chapter 2).