Erica,
I don't think there was a law about it, but as a general rule I would be surprised to find a boy less than 16 traveling by himself. Culturally at the time, 16 was a significant age. Both in England and America it was the start of period we might call probationary adulthood. Sort of like 18 used to be in our lifetimes. You'd get some rights at 16, but not all of them. For example, 16 was the age for entry into military service, but a boy of that time couldn't own land or sign contracts until he was 21. At about 14, a boy would normally be apprenticed so that his term would end in 7 years when he was 21. At 16, a boy could easily travel around alone and find work to support himself, albeit with some suspicion that he might be a runaway apprentice.
We know from detailed analyses of English households of this period that "servants" were often some kind of impoverished or orphaned "shirt-tail" relatives. A man might take on the orphaned son of his wife's sister's late husband's cousin, because the kid had to go somewhere and the family network decided this household was the logical place.
The category of servant could also include apprentices. It would be rare at this time, I think, for a family to travel with an actual indentured servant. The whole idea of indentured servants was that they were working off their passage to America. If someone were an actual servant or apprentice the master would naturally have paid for the passage. No doubt, though, that many expected an extra term of work in America as payment.
Then too, many scholars think that the term servant as applied to many of the people on the surviving passenger lists might have been just a convenient shorthand for a someone traveling with the family, for whom the master was taking responsibility. Could be a friend or distant relative. Could be an orphaned child put in the care of this particular family. But, probably some kind of connection.
I have an ancestor who was sent to America as a young girl by her Puritan father to get her out of the way of the English Civil War. She went with a distant cousin who was going. It's all very well documented. I'm sure her father meant to bring her back, but he died. Her brother was apparently not anxious to split the estate with her, so he never quite sent the money for her passage back. She stayed and married. I've always imagined that might have happened in many more cases than we know or can prove.