I think finding proof of this Scottish connection is impossible.
This old family tale is typical of the stories so many Swedish noble families invented back then. They were newly ennobled, and wanted to give an impression they had old noble ancestry.
The suggested birth year 1601, based on the family tale, "came to Sweden 14 years old in 1615", does not add up well with his first son Jöns born 1616. (Szabad 2008 and Örnberg 1884)
Szabad (2008) and Örnberg (1884) have "ca 1580"/"1580s" as birth year for Anders Hansson. I think we should go with that.
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Szabad (2008) vol.1 p.28-29 "Adelsköld nr 2029" Tab.1:
"Anders Hansson, f. omkr. 1580, † före 1656. Ytterligare barn: Jöns Andersson, f. 1616, † 1671, komminister i Helsingborg, kyrkoherde i Jämshög, g. 1:o 1647 m. Elene Olsdotter, † 1650, g. 2:o 1652 m. Malena Lauridsdotter i hennes 1:a gifte (g. 2:o m. Jöns Sorbonius, prost i Jämshög). Hans Andersson, f. 1625, † 1674, prost i Vinslöv, g. m. Eline Pedersdotter i hennes 1:a gifte (g. 2:o m. Peter Norbergh, rådman i Kristianstad), f. 1625. "
"[SBL:1 s78; ÖÄ:10 s474ff; ShF:1987:4 s18]"
The abbreviated references above that Szabad (2008) refers to for Tab.1:
SBL Svenskt biografiskt lexikon (Stockholm 1918-) Vol.1 "Adelsköld"
https://sok.riksarkivet.se/sbl/Presentation.aspx?id=5527
https://sok.riksarkivet.se/sbl/Presentation.aspx?id=5529
ÖÄ Örnberg, Victor: Svenska ättartal 1884, vol.10 p.478
"Anders Hansson, antagligen född på 1580-talet, levde än på 1620-talet"
https://www.alvin-portal.org/alvin/imageViewer.jsf?dsId=ATTACHMENT-0496&pid=alvin-record:517981
ShF Släkthistoriskt Forum (1987) nr.4 p.18
https://www.rotter.se/premium/shf-arkiv
(not free)
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Szabad (2008) and Örnberg (1884) have a lot of descendants to Anders Hansson that is not listed in Elgenstierna/Adelsvapen.
Szabad (2008) has a lot of corrections to Elgenstierna/Adelsvapen.