Why did Jethro help Eb in Chapter 9 of Across Five Aprils?

Jethro helps Eb in Chapter 9 because, when all is said and done, he would not be able to live with himself if he does not help someone who is so in need.

In his young life, Jethro "had never been faced with the responsibility of making a fearful decision like the one confronting him".  He is completely aware of the consequences he and his family might have to face at the hands of the law if he helps Eb, who is a deserter.  Also, he recognizes within himself a bit of resentment towards Eb, who has chosen to run while his brothers are sticking it out.  On the other hand, he puts himself in Eb's shoes, wondering how he would react if he were as "sick and scared and hopeless" as he knows Eb is, and he considers that Eb has acted heroically for two long years, and that to ask him to do so indefinitely might be too much to ask.  Mainly, though, Jethro wonders how it would feel to know that he was the one that sent his cousin to his death.

Jethro considers asking his Pa what he should do, but recognizes that his Pa would "be caught in the same trap (he's) in now...he'd put him in the spot where any way he decided would be bad - hurtful to a man's conscience".  Jethro understands that his decision about whether or not to help Eb is one he has to make himself, and that there are no easy answers.  In the end, like President Lincoln himself who grants amnesty to deserters like Eb who wish to return to their regiments, he makes his choice to the best of his ability after examining the situation from all sides, choosing to err, if he must, "on the side of mercy" (Chapter 9).

Why did Jethro get in trouble for aiding Eb in Across Five Aprils?

Jethro was in trouble for helping Eb in Chapter 9 because Eb was a deserter from the Union Army, and according to the law, anyone hiding or otherwise helping deserters were in direct defiance of the law as well.

In the winter of 1863, desertion was a big problem in the Union Army.  The war had dragged on for going on three years, and the soldiers who had survived to that point were tired, traumatized, and demoralized.  Deserters by the score wandered the land; desperate, they often formed bands and terrorized the citizenry around them.  It was the job of the representatives of the Federal Registrars to hunt down the deserters and bring them in for punishment.

One night in February of 1863, representatives of the Federal Registrars came to the Creighton home looking for Eb.  After searching the house and farm to no avail, they warned the family in no uncertain terms that shielding a deserter was a serious crime.  Although the punishment for the crime was not specified by the men, their message was clear.  They told Jethro that if Eb should turn up, he was to "get word to the Office of the Federal Registrars in Chicago right away, telling them where the man is or expects to be".  If Eb should show up and Jethro were not to report him, he and his family would "be up to (their) necks in trouble" (Chapter 9).