This was 1864, not 1870. And (to be historical correct) this part of today Hamburg, "Altona" was a part of the duchy Holstein. The King of Denmark was also Duke of Holstein and Lauenburg (both parts of the German federation) and Duke of Schleswig. From 1864 to 1866 Austria and Prussia ruled Schleswig-Holstein together and since 1886 to 1945 it was a prussian province.
Altona was an independent city til 1937 and since that year a part of Hamburg.
Birth place for a child would have been "Altona" or one of the quarters.
But I doubt a little bit (because of the name) that they are originally from Hamburg. Maybe they lived before in an area with mixed german - polish population. You should try find them in the Hamburg passenger lists. There is said, where their home-town was.
Maybe the name Boutz was originally Bautz and the other name sound a little polish. So you should try different versions.