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For exactly $4$ boys, and $4$ girls I got $70/256$, but I'm not sure how at least $4$ boys change the scenario here.

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    you got the probability of 4 boys. Now add in the probability of 5 boys, the probability of 6 boys, 7 and 8. In all those events (and only those events) there are at least 4 boys – 2017-02-20

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You have the probability of exactly 4 boys. If that doesn't happen, then the probability of more than four is equal to the probability of less than four. So divide the probability of "not four" by two and add it to "exactly four". I get

$$\frac{1}{2}\frac{256-70}{256}+\frac{70}{256} = \frac{163}{256}$$