Today we have more resources available to us than Converse had in 1905. I found Church of Boston and various town records listing Edward Allen, Boston Merchant, his marriage, birth of his children, numerous deed references to his property and even his death. I have to say, as Sheila suspected, the younger Edward Allen, Boston Merchant, is definitely not the same person as Edward Allen of Ipswich.
Edward Allen of Boston married Martha Waye[Way] March 7, 1652, Boston, performed by Gov. Thomas Dudley, First Church of Boston.
Children born Boston, Suffolk Co., MA: Sarah Allen b 1653 John Allenb 1657 (died young) Edward Allen b 1661 Eleazer Allen b 1664 Martha Allenb 1666 Mary Allenb 1668 Elizabeth b 1672
2nd wife Lydia (possibly Ludkin) John b 1673
It was not always clear in the church records as to birth vs baptism dates. Edward lived on Hudson Lane, Boston and was a Tailor. He was clearly a member of the First Church of Boston. He was married there. All of his children were baptized there. Edward died May 31, 1693. In short this Edward Allen, Boston Merchant, had a well documented life and nothing about it matches Edward Allen of Ipswich. I went through all of those records and found no other adult Edward Allen in Essex or Suffolk Counties during those years. Of course, this Edward had to name a son Edward (1661), just like Edward of Ipswich had to name a son Edward (1662).
During the same years, an older and younger James Allen were also closely associated with the First Church of Boston. It is possible the older James was this Edward's father, who would have then been a brother to Edward Allen of Dedham.
I didn't find anything that would even remotely connect these Allen's with my Edward Allen of Ipswich. Well... they were both Puritans, but then nearly everyone in Mass. Bay Colony was a Puritan. So, we still have no idea if Edward of Ipswich was born in America to yet unknown parents or came over shortly before marrying Sarah Kimball in 1658 when he first shows up in a record.