Since the square root of $n$ is between 17 and 18, then $n$ is between $17^2 = 289$ and $18^2 = 324$. Since $n$ is a multiple of 7, we need to count the number of multiples of 7 between 289 and 324. Since $41 \times 7 = 287$ and $42 \times 7 = 294$, then 294 is the smallest multiple of 7 larger than 289. Since $46 \times 7 = 322$ and $47 \times 7 = 329$, then 322 is the largest multiple of 7 smaller than 324. This means that the multiples of 7 between 289 and 324 are $42 \times 7 = 294, 43 \times 7 = 301, 44 \times 7 = 308, 45 \times 7 = 315$, and $46 \times 7 = 322$, of which there are 5. (We note that we could have determined that there were 5 such multiples by calculating $46 - 42 + 1$ which equals 5.) Therefore, there are 5 possible values for $n$.