Hi,
I don't know if the information you seek is going to be available on the internet. Presumably at some time, the family must have had some form of naturalisation papers for Jacob and his daughter Sarah, but I would not know if it is possible to obtain copies or whom you would currently approach.
As an aside, one of my wife's family was a naturalised German and was granted a Certificate of Naturalisation from the Home Office in London in 1879 under the Naturalization Act of 1870 - a - "Certificate of Naturalization to an Alien" was issued and as part of this, they were required to sign an oath of allegiance. Fortunately from our point of view, the documentation was retained by the family.
Also, have you thought about looking/contacting the National Archives at Boston to see if they have any emigration records for travellers from Boston to Liverpool?
Alan