I found this little bit of info in John Farmer's 1816 book, "Memoir of Billerica, Massachusetts".  It says that the first European inhabitants had originally called the town of Billerica, Shawshin, probably after the river of the same name, Shawshin (which we now call Shawsheen). 

To the Indians that were already here, they named it Shawshinock or Souhegenock which they in turn called the Shawsheen River. 

Then the name was changed to Billericay for Billericay, England,  the town for which some of the first settlers came from.  And the last and final name it became is the present name it has today - Billerica.

As I was browsing about the Internet and came across two Billericay, England websites, it lists on their histories for the naming of their town - Billerica and THEN Billericay!  The opposite in naming order for  its town to its American counterpart!

Who can figure?

-Binny