And despite Uncle Richard's success - and the fact that a distant cousin was Italian superstar Marcello Mastroianni - Masterson says he wouldn't have believed you could make a career out of performing or even where to start, anyway.
But that road trip changed Masterson's life.
He is now one of the world's most successful solo performers and an award-winning independent theatre producer, having presented more than 150 shows in 22 years at the Edinburgh Fringe Festival.
It took a little while for him to get there. Masterson moved to the United States to work in the early 1980s and saw Burton just a handful of times before his death in 1984. It wasn't until then that he started acting classes, taken by a girlfriend who was impressed by his theatrical lineage. Eventually, Masterson completed training at the London Academy of Drama and has been pretty much on the road ever since, touring or producing.
Now he returns to New Zealand to perform two solo shows: Thomas' acclaimed Under Milk Wood and a comedic Shylock, based on Shakespeare's notorious character. Under Milk Wood was one of Masterson's first solo shows created to pay homage not just to Thomas but to Uncle Richard and the memory of their road trip. Burton made a well-known BBC radio recording of Under Milk Wood in 1954.
In his version, Masterson recreates 69 characters in Under Milk Wood, set during the course of a day in a small seaside village. It's a piece he has performed more than 2000 times during two decades. The second work, Shylock, explores the tragic, tempestuous fate of one of literature's most famous Jewish figures - the conniving moneylender of Shakespeare's Merchant of Venice - delving into stereotypes and identity as it does so.
"I've gained a reputation for making and performing solo shows, so people bring ideas to me."
While his family connections are well-known, Masterson's impact on New Zealand theatre is less talked about but he spent much of the late 1990s helping local actors take their shows to the Edinburgh Fringe Festival.
"I was in New Zealand in 1997 and I saw Tim Balme's show [The Ballard of Jimmy Costello] and told him he should take it to Edinburgh. Tim said he would if he could find someone who could show him the ropes so I said, 'I'll do it'."
The following year, he produced Gary Henderson's Skin Tight in Edinburgh - it won a Fringe First Award - and, in '99, he was back (after a working honeymoon in Auckland) in Scotland with a new NZ theatre company called Indian Ink and its debut play, Krishnan's Dairy. That helped the company become one of NZ's most successful, forging them connections around the world.
"Toa Fraser's Number Two, with Madelaine Sami, came in 2000 and I remember Toa at dinner one night talking about he was going to try writing a film script from the story..."
Back in New Zealand for the first time in several years, Masterson has headed further south than even before, performing in Dunedin, and is looking forward to catching up with his many friends in Auckland.
Under Milk Wood plays on Thursday and Friday at 7pm and Saturday at 2pm; Shylock is on Wednesday and Saturday at 7pm.
What: Under Milk Wood and Shylock starring Guy Masterson
Where & when: Herald Theatre, October 19 - 22