Custer specifically? Nathaniel Philbrook's book on Little Bighorn makes no mention of whether or not Lt. Col. Custer (General was a brevet rank he earned during the Civil War, but he reverted to his Regular Army rank after the war) did so, but you are correct that dead bodies will often do 'doo-doo'.As for non-dead bodies, in battle, many people will crap themselves - According to Lt. Col. Grossman's On Combat, its about 25 percent (I say about, as I don't have the book on hand just now, but I think thats the number he gives), but many of those occurrences aren't out of fear, but out of a physiological reaction by the body to such stressful situations. Its like a fight or flight reaction, where the body needs to concentrate wholly on the situation at hand, not on what is in your lower intestines. Same for urine. Many soldiers don't even realize they did until a good deal after it happened. Anyways though, that is a reaction that occurs very early in the fight for the most part.

Anyways though, as I said, I've read nothing to indicate one way or the other. The only witnesses were the native Americans, and their accounts are very conflicting. We aren't sure who killed him or when exactly he died in the course of the battle (although most claims put it near the end, at least some claim he fell early in the fight). Based on the wounds he sustained, it is generally thought he died pretty quickly, so he wouldn't have had time to crap himself while lying their wounded either.

So anyways, to settle your bet. We don't know if he crapped himself, but if he did, it probably was after death. If it was before he died, it was most likely an physiological reaction, not out of fear, as whatever his faults, he was pretty roundly acknowledged for his bravery.