Have you looked in the Ellis Island web site for the name BEGA or was it maybe BIGA?
The name Biga does list Humenne, Nova Sedlica.
The name Begova (female) lists in Parihuzovce, Humenne, Bratislava and Osadne. The name Bega lists in Humenne, Cabalovce, Michalovce, Vrutky, Vydran, Medzilaborce.
The Slovak phone book is at www.zoznamst.sk
The LDS-Mormons have filmed the church records but you do need a name of a town to order the films and gather family information.
If he was naturalized citizen or had social security numbers than you can request those papers and find the name of the town that he came from.
The old 1910 Hungarian megye map is at http://lazarus.elte.hu/hun/maps/1910www.slovakemb.comwww.iarelative.comwww.hosmanek.com/slovakwww.village.skwww.slovensko.comwww.our-slovakia.comhttp://mapy.atlas.sk