If my grandmother (Anna Marie Bender) and my mother (Kati, age 3) listed this town on ship's immigration information from Liverpool to New York, where might they have boarded the ship from Croatia to Liverpool?

Is there a reason you posted this on Bosnia Herzegovina board – since they as you said were from Croatia?

Palešnik is in Central Croatia – the closest port would be Rijeka (back than known as Fiume as well), but it really means nothing, Croats emigrated through Le Havre, Trieste, Hamburg, Bremen, Rotterdam etc.

Most likely they took a train to France and then to England.

Robert

Yes because at the time I did not know the exact location and now I do and it is not Hercecovina but Herzegovac.

palesnik is a vilige near town garesnica, and all records is save in this town
http://udubbz.hr/mat-ured-garesnica/
in this link is only phone, but no e-mail adress, sorry