https://familysearch.org/pal:/MM9.1.1/K4R8-LNB

Name: Anthony Zbikowski
Titles and Terms:
Event Type: Census
Event Date: 1940
Event Place: Ward 2, Bayonne, Bayonne City, Hudson, New Jersey, United States
Gender: Male
Age: 73
Marital Status: Married
Race (Original): White
Race: White
Relationship to Head of Household (Original): Head
Relationship to Head of Household: Head
Birthplace: Poland
Birth Year (Estimated): 1867
Last Place of Residence: Same House
District: 9-32
Family Number: 5
Sheet Number and Letter: 7B
Line Number: 77
Affiliate Publication Number: T627
Affiliate Film Number: 2345
Digital Folder Number: 005462003
Image Number: 00188
Household Gender Age Birthplace
Head Anthony Zbikowski M 73 Poland
Wife Anna Zbikowski F 55 Poland
Son-in-law John Bronikowski M 32 New Jersey
Daughter Sophie Bronikowski F 28 New Jersey
Grandson John Bronikowski M 3 New Jersey

Is this your grandparents in 1940 census?
Both ancestors have Pa-first paper. They started their naturalization process. You might want to check into that. If they did not naturalized than an Alien Registration Form they had to fill out. There would be a place of birth.

http://www.archives.gov/research/naturalization/naturalizati...
http://www.genealogybranches.com/naturalization.html

By any chance ancestor's death certificate has social security numbers written on there?
http://www.moikrewni.pl/mapa/kompletny/bronikowski.html