Step 1: Identify the total number of integers

The integers are from $1$ to $70$ inclusive, so the total count of numbers is $70$.

$\text{Total numbers} = 70$

Step 2: Count numbers with unit digit 1 or 7

In every group of 10 numbers (e.g., 1-10, 11-20), exactly two numbers have 1 or 7 in their unit's place (e.g., 1, 7; 11, 17; etc.). Across 7 blocks of 10 from 1 to 70, the number of such integers is $7 \times 2 = 14$.

$\text{Favorable count} = 7 \times 2 = 14$

Step 3: Calculate the percentage

Divide the favorable count by the total count and multiply by $100$: $\frac{14}{70} \times 100 = \frac{1}{5} \times 100 = 20\%$

$\text{Percentage} = \frac{14}{70} \times 100 = 20\%$