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Of course he can fight it. First he can get out of jail by posting bail. The he can defend himself in court against a charge of burglary - if he can't afford a lawyer the court will appoint one at no charge. Texas law defines "burglary" like this:
§ 30.02. BURGLARY. (a) A person commits an offense if, without the effective consent of the owner, the person:(1) enters a habitation, or a building (or any portion of a building) not then open to the public, with intent to commit a felony, theft, or an assault; or(2) remains concealed, with intent to commit a felony, theft, or an assault, in a building or habitation; or(3) enters a building or habitation and commits or attempts to commit a felony, theft, or an assault.(b) For purposes of this section, "enter" means to intrude: (1) any part of the body; or (2) any physical object connected with the body.
Since your boyfriend's only "intent" was to drop his dauhter off, he did not commit "burglary". This is a non-starter for the prosecution and your boyfriends lawyer will see that in a minute!
All the best,