Looking it in factors. 

(x-1)(x^2+x+1)(y-1)(y^2+y+1) = z^2

The bold expressions are either prime numbers or their factors have "big" prime numbers (if comparing to x or y). When the values of x and y increases some big prime numbers would get involved. It would be very difficult to have these primes twice: sometimes, but is rare, one of the bold expression have a factor repeated (as in Charlie's solutions), in this case, the chances of solution increases. Also when both bold expressions have some common factor the chances of solution are higher. 

Other way to look for solutions is seeing that (y^3-1) is a square multiple of (x^3-1) [At least in the solutions]. So a good strategy for computer is take a value of (x^3-1) and have the product with all the integer squares from 1 to 10^9 (p.es) and see if one of this match any (y^3-1). I've done it with Excel for x=5 to 11 and didn't find anything with the first 50000 squares. 

But it wouldn't surprise me if there is somewhere a theorem saying that no more solutions are possible.