Does anybody know what is the origins of the Szajkowski surname?
Many names ending in -ski are of Polish origin, but they're also common in Slovak, and they occur (in some spelling or other) in all of the Slavic languages. The suffix means "of, from, deriving from or belonging to", and as such can mark both locative (place-derived) and patronymic (parent-derived) names.
Szajkó is the Slavic-derived Hungarian name for the Eurasian jay. (It has the vowels transposed from the Polish and Slovak forms of the word: sójka, sojka.) As the Latin name of the species (Garrulus glandarius) implies, jays are mimics, and can be taught to "speak", so the byname Szajkó could originally have been bestowed on a talkative person, or someone who spoke loudly, or who habitually repeated phrases (in inappropriate contexts).
However, if the first part is the bird-name, then the name as a whole is a combination of orthographies (spelling systems): Hungarian for the Sz-, and Polish (or possibly German) for the -owski. (Hungarian uses s versus sz exactly backwards from Polish: in Hungarian, s is /sh/ as in "sham" and sz is /s/ as in "Sam", while in Polish, s is /s/ and sz is /sh/.) Therefore, I wonder if Szajkowski could be a slight Americanization of a different Polish bird-derived surname, Czajkowski (=Tchaikovsky, like the composer), which indicates association with one of several places named after the lap-wing or plover (Polish czajka).
If the question was for practical application, then a quick search of historical records on FamilySearch points largely to Poland. On the web interface search on the surname with exact match, and look in the left sidebar in the Filters list under Birthplace. Within Poland, there is no particular concentration reflected in records on FamilySearch.