According to the 1840 Tippecanoe County, IN Lauramee Twp census, there was a John Beck Boys under 5=1 Boys 5-10= 2 Boys 10-15-2 Men 30-40=1 Girls under 5=1 Women 30-40=1 So therefore it appears that John A. Beck was the son of a John Beck.
1850 Tippecanoe County, IN Lauramee Twp there is a 14 yr old John Buch living with a Thomas Jones family. No John Beck's in 1850, but there is an Elizabeth Beck, with children, possibly his mother.Elizabeth is born in Virginia, the children all born in IndianaThey are a few houses from the Thomas Jones household.
1860 Tippecanoe County, IN Lauramee Twp there is a 24yr old John Beck living with the James Oglebay family.
1870 Tippecanoe County, In Wea Twp, Town of Romney John and Sarah along with a son, Harry.
1880Indiana > Tippecanoe > Lauramie > District 51 John, Sarah, Harry, Elizabeth, Hettie and Rebecca
Hopefully this will give you something to work on. I am not related, but from Tippecanoe County originally. just trying to help out a fellow genealogist