Yes, that seems correct! Your grandfather's birth name was Jöns. I can see why John would be easier in the US. The other children listed are daughter Ingar (1869), and sons Måns (1873), Håkan (1875), and Elof (1877). All born in Billeberga parish. The family lived in the village of Tostarp 1876-1884 (same parish). Before that, they lived in Billeberga.
Also, Per was married previously. With his first wife, Ingar Persdotter, he had a son, who was also named Jöns. This Jöns passed away after only 2 days and Ingar died a week later, from complications after the birth. Here they are in the parish death records (row numbers 9 and 12): Billeberga kyrkoarkiv, Födelse- och dopböcker, SE/LLA/13023/C I/6 (1835-1864), bildid: C0063837_00099
https://sok.riksarkivet.se/bildvisning/C0063837_00099 and in the household records: Billeberga kyrkoarkiv, Husförhörslängder, SE/LLA/13023/A I/5 (1857-1866), bildid: C0063825_00017https://sok.riksarkivet.se/bildvisning/C0063825_00017. So your grandfather was named for his older brother and it seems his sister was named for their father's first wife.