(Note: For international students)
If someone has a gap of more than 3 years after obtaining a masters degree, are they eligible for a PhD in the USA?
There is no restriction that a PhD should be done right after a Master's degree.
For instance, I knew a returning PhD student of age >50 at a major US university.
Depending on what the candidate did in the three years in between, snatching a precious place in a PhD program may be either more difficult or easier.
Edit: I didn't exactly answer the question before. The short answer is that there is no such eligibility requirement. Long, original answer below.
As with any job application, a gap is only as bad as you describe it. You can "sell" yourself effectively by describing that gap as positively as possible-- whether you were working professionally, raising a kid, traveling, helping out your family, etc. And quite often, the life experience you gain by having a break from schooling is a huge asset when you enter a grueling graduate program.
I don't have data to back it up (this is close enough), but my perception is that the average age of PhD students in the U.S. has gone up over the past few decades, because people often need to work a "real" job, save up money, and pay off loans before taking on the job of an underpaid grad student with excessive hours. So three years between a masters and PhD seems about right.