I think you want to just create a new random variable, also with a binomial distribution, but now the underlying $p$ is given by the probability of hatching, i.e.
$p=Pr[\mbox{egg produced}]Pr[\mbox{egg hatches}]=\left(\dfrac 3 4\right)\left(\dfrac 7 {10}\right)=\dfrac{21}{40}$
As before $n=8$
Now just evaluate the CDF of this new binomial distribution at $k=6$ and the probability you want will be 1 minus this value.