I was wondering if you found the information you were looking for? I stumbled across your posting on the message board while helping my Aunt research our family tree. While searching for information about my 2xGreat grandfather (I believe his mother Emma married a John Mycock/Maycock after he was born) I might have inadvertently found your John Mycock information (spelled Meycock, b. 1864) on the 1900 United States Federal Census in Trenton,NJ, He was listed with a wife Elizabeth Mycock, b. 1853 and three sons with the same names you were looking for John, b.1888, William, b.1889, and Earl, b.1895.
I also found a John Mycock, b.1866, listed on the 1881 England Census, in Burslem, Staffordshire, who might be the same person. I think my 2xGreatgrandfater also listed on this same 1881 census was a stepbrother to this John Mycock . I believe my 2xGreatgrandfather immigrated to the US about 1885 (I have yet to verify this information) however he did settle in Trenton,NJ (he worked for the Potteries) so it sort of makes sense that this might be the same John Mycock b .1864/1866.
I know it has been 3 years since you asked for this information so you might have already checked this lead out, but if you haven’t I hope this information might help you.