Well, that would definitely make it easier for younger kids to understand why everyone on both sides of the conflict would be so opposed to them getting together.
It certainly would've spiced up that silly version from the mid-90's that starred the young Leo DiCaprio!
Although, I don't think making the characters gay at this point would be modern enough--it probably wound't even register to most of these kids as being odd (or dare I say queer?). One of them (or both of them) would need to be trans--but in the end, all you'd really have is just a weird mashup of Shakespeare and The Crying Game. Which would still place that story firmly into the 90's!
Honestly, without throwing in blatant references to modern technology, is there any possible way to make that story more modern that they were capable of back in the 90's? I'm starting to think we've reached peak Shakespeare!