> I'm wondering if there is any relation between the Hamborgar and the "Hambo" dance that is commonly done at breaks in contradances.

In a word, no. Here's one version of Hamborgar, which is similar to a polka, in 2/4 or 4/4 time. Hambo is a derivative of Swedish polska, in 3/4 time. There is a fairly standard version of Hambo that is widely done, though there are also other versions that were collected from various places.