Nothing turned up in the passenger lists for me, either.
If you don't have his parents' names, you might try getting his death certificate. Check here:
http://www.ct.gov/dph/cwp/view.asp?a=3132&q=390664 Also, you might want to try to get his obituary. Occasionally, good details like parents' names, when they married, or when the person came to the US might pop up. You could try contacting the local library in Rockville/Vernon or look on raogk.com for a volunteer.
I saw in the 1908 Rockville directory that James J. Burns removed to Clinton, Massachusetts. If you want to figure out where they were in New York when the two older kids were born, he's probably be the easier one to follow, since the other was female and may have married. His place of birth might show up on some vital record or draft registration card.
1910 Census, Clinton, Worcester Co., Mass., 4 Henry St.
Burns, James J.S., age 25, married once, number of years of present marriage - 0, born NY, parents born Ireland, occupation - cloth inspector, Worsted Mill
wife Mabel R., age 26, married once, number of years of present marriage - 0, mother of 0 children, born Connecticut, father born Connecticut, mother born Canada - French, no occupation
Also living with them are sisters-in-law Grace M. and Hasel M. Falber.
1920 Census, Coventry, Kent Co., Rhode Island, 19 Main Street
Burns, James, age 35, married, born New York, parents born Ireland
I'm having a hard time getting the image open now, but he's living with wife Mabel, age 35, son James, age 9, daughter Norma, age 1 year and 9 months, and sister-in-law Grace Falter or Faller, age 25.
1930 Census, Coventry, Kent Co., Rhode Island, 70 Maple Station
Burns, James J., age 45, married for 25 years, born NY, parents born Ireland, overseer at Worsted Mill, not a veteran
wife Mabel, age 46, married for 25 years, born Connecticut, father born Connecticut, mother born Canada - French, no occupation
son James E., age 19, single, born Massachusetts, father born NY, mother born Conn., no occupation
daughter Norma C., age 10, single, born Rhode Island
son Russell, age 9, single, born Rhode Island
daughter Marion, age 5, single, born Rhode Island
son Donald, age 2, single, born Rhode Island