Rick, let's cut to the chase on your query.The standard genealogy for the three northern Churchill families in America is "Churchill Family in America," (CFA) published in 1904.While it has errors and omissions, it usually is a sound starting point because the compilers checked and cross-checked and were as comprehensive as they could be.Internet postings are, of course, not vetted or reviewed and thus are only as good as the poster.
While it would have been useful if you could have provided more information, you did provide the key point that the father of the Eunice you are looking for was a Jr.Although the Massachusetts family has numerous Nathaniels, only one is a Nathaniel, Jr. who has a daughter named Eunice.That is the Nathaniel, Jr., shown by CFA as married to a Eunice Kinney, about whom George Churchill posted.This Nathaniel, Jr. is number 169 in CFA.
There is one other Nathaniel, Jr., no. 102 in CFA, on whom there is no further record after his birth.There also is a Nathaniel III, no. 400 in CFA, who had a daughter Eunice who married an Amos Fox.Since you do not mention the Fox family name in your posting, I would assume this is not your Eunice.
The Connecticut Churchills are an even simpler matter.With one exception, the Nathaniel Churchills in the Connecticut family run in a single line, Nathaniel I through Nathaniel V.I find no record of any of these five either marrying a Eunice or having a daughter named Eunice.There is in addition one other Nathaniel who was not a Jr. and on whom there is no further information after his birth.
In short, it would appear that you do not need to search beyond the Nathaniel Churchill, Jr. about whom George posted. If for any reason that does not check out, the likelihood is that your Eunice does not belong to one of the three northern families but is either of one of the Virginia families or of a family that immigrated later to the U.S. or Canada.