The St. Louis to KC Amtrak is very bike-friendly because of the Katy Trail. The bikes are leaned up against the front of the car, where they take out a seat to leave room. You do have to pay extra and reserve a spot (I think they only allow 3?) The only problem we had was it was a quick train stop and we had to scramble to roll the two bikes up onto the train (the conductor did help). However, the Katy forum has stories of people being denied, even though they prepaid a spot.
Call the local train station and try to get specifics. A box may be safer, but it you have to transfer it yourself, this could be more difficult. Is there a local bike forum where you could look for advice? Maybe call a bike shop in NY and ask for tips.
"Well-behaved women seldom make history." --Laurel Thatcher Ulrich
'09 Trek WSD 2.1 with a Brooks B-68 saddle
'11 Trek WSD Madone 5.2 with Brooks B-17