jmc

$343=7^3=1000_7$, so the first 343 natural numbers in base 7 are $1_7,2_7,\dots 1000_7$. Any number in this list that neither includes 4 or 5 only includes the digits 0, 1, 2, 3, and 6. If we replace 6 with 4, these have the same decimal expansions as the integers in base 5. Since there are $5^3=125$ positive integers less than or equal to $1000_5$, there are 125 integers less than or equal to $1000_7$ that contain no 4's or 5's in base 7, which means there are $343-125=\boxed{218}$ integers that include a 4 or a 5.