SINKOVITS is most likely a Germanized variant of the Croatian name SINKOVIĆ.
There is a place in SE Austria (which was part of the Kingdom of Hungary until after WW I) called Burgenland. You will find people researching SINKOVITS in areas where Croatians were known to have settled. This group arrived after the invastion of the Ottoman Turks of their Croatian homeland. http://www.the-burgenland-bunch.org/Surnames/surnames.html
Also you will find several interesting articles about Croat migration to Burgenland (which began in the 1500s)http://www.the-burgenland-bunch.org/Newsletter/NewsletterInd...
Other links of interesthttp://boards.ancestry.com/localities.ceeurope.austria.Prov....
A few years ago I helped the young man who posted the original message below and found that part of his family was from Croatia proper and... his SINKOVITS family were Burgenland Croats!http://boards.ancestry.com/localities.ceeurope.austria.Prov....
Familysearch.org is another good tool, where you will find SINKOVTIS (Germanized), SINKOVICS (Hungarianized) and SINKVOICH (Americanized) but in any case you will need to know his EXACT birthplace to do any overseas research.
Mathias arrived under the Displaced Person's Act which allowed about a half million displaced people to come to the US. Note that he and the family was listed as Stateless on the airplane manifest, which means the likely came from a DP Camp (these camps existed into the early 1960s!).
Did he become a US Citizen? If so then his Declaration and/or Petition will give his birthplace. Or if this is your Mathias Social Security Death Index, below then his application which will cost you $27 (since we have his SS number) will have his birthplace. Be aware that in rare cases it will only give the country, but in most cases the actuall birth town is given.
Name: Mathias Sinkovits
Last Residence: 60618 Chicago, Cook, Illinois, United States of America
Born: 28 Aug 1925
Died: Nov 1989
State (Year) SSN issued: New York (1956-1957)
I would begin by ordering his SS App and... hope it has his town of birth. Below is a link to the SSA and info re ordering recordshttp://ssa-custhelp.ssa.gov/app/answers/detail/a_id/147/rela...
I hope this will at least help you get started.
Croatian Heritage Museum