Hello Ben,
From the little I know, R' Yonatan Eibeschutz seems to be one of the luminaries of a particularly fascinating period of Rabbinical history. His ascending lineage seems somewhat unknown (as far as I am aware) and so it would be worthwhile I think to try to connect his line to some other rabbinical lineages, which can be done using yDNA. As you may know, the thing about yDNA is that it flows only from father to son, which means that any interruption of that sequence means the original yDNA signal is lost. Looking at your tree, while you are certainly a 7x ggs of R' Eibeschutz, your yDNA is that of the Frydman line so would not help in the search for his connection to other rabbinical lines.
I will explain how you can find out more about the living descendants: first go to the DNA tab on his profile, then under YDNA select 'View a List of Living People...'. This will bring you to the list of 10 possible yDNA descendants alive today. From here you have to click on 'View Tree' to see how each person connects to the line. If you want to try to contact anyone you can use 'Contact Profile Manager' next to the name.
Genealogical research is very much a percentage game. A good percentage of leads go nowhere, but every once in a while something interesting turns up.
Good luck and please keep me posted if you happen to locate a potential ydna provider. In certain situations it is possible to get some funding for the labwork.
Be Well,
Adam