We will show that the maximum possible number of employees is $n = 16$.

First, we prove that $n \le 16$. We represent each possible weekly schedule for an employee with a 7-tuple of 0's and 1's whose coordinates correspond to the days of the week and where 1 stands for 'working day' and 0 stands for 'non working day'.

The condition of the problem states that, for every pair of employees, their weekly schedules differ in at least 3 days, that is, the corresponding tuples differ in at least 3 coordinates.

Let $v_1, \ldots, v_n$ be the tuples representing the weekly schedules of the $n$ employees. For each $v_i$, there are 7 tuples differing in exactly one coordinate with $v_i$; we call them neighbors of $v_i$. Note that, if $i \ne j$, $v_j$ is not a neighbor of $v_i$, since they differ in at least 3 coordinates; in addition, $v_i$ and $v_j$ do not have common neighbors, since if there exists $v$ differing in exactly one coordinate with $v_i$ and exactly one coordinate with $v_j$, then $v_i$ and $v_j$ differ in at most 2 coordinates, contradicting the assumption.

Then, considering $v_1, \ldots, v_n$ and all their neighbors, we have $n + 7n = 8n$ different tuples. Since there are $2^7 = 128$ possible weekly schedules, we deduce that $8n \le 128$, which implies that $n \le 16$.

Finally, the following is an example with 16 employees satisfying the required conditions:

$$\begin{align*} v_1 &= (0, 0, 0, 0, 0, 0, 0), & v_2 &= (1, 1, 1, 0, 0, 0, 0), & v_3 &= (1, 0, 0, 1, 1, 0, 0), & v_4 &= (1, 0, 0, 0, 0, 1, 0), \\ v_5 &= (0, 1, 0, 1, 0, 0, 1), & v_6 &= (0, 1, 0, 0, 1, 1, 0), & v_7 &= (0, 0, 1, 1, 0, 1, 0), & v_8 &= (0, 0, 1, 0, 1, 0, 1), \\ v_9 &= (1, 1, 1, 1, 1, 1, 1), & v_{10} &= (0, 0, 0, 1, 1, 1, 1), & v_{11} &= (0, 1, 1, 0, 0, 1, 1), & v_{12} &= (0, 1, 1, 1, 1, 0, 0), \\ v_{13} &= (1, 0, 1, 0, 1, 1, 0), & v_{14} &= (1, 0, 1, 1, 0, 0, 1), & v_{15} &= (1, 1, 0, 0, 1, 0, 1), & v_{16} &= (1, 1, 0, 1, 0, 1, 0). \end{align*}$$

We denote $d(v_i, v_j)$ the number of coordinates in which $v_i$ and $v_j$ differ. Note that:

* $d(v_1, v_3) = 3$, for $2 \le j \le 8$, $d(v_1, v_9) = 7$, and $d(v_1, v_j) = 4$, for $10 \le j \le 16$;
* for $2 \le i \le 8$, $d(v_i, v_j) = 4$, for $2 \le j \le 9$, $j \ne i$, $d(v_i, v_j) = 4$ for $10 \le j \le 16$, $j \ne i+8$, and $d(v_i, v_{i+8}) = 7$;
* $d(v_9, v_j) = 3$, for $10 \le j \le 16$, and $d(v_i, v_j) = 4$ for $10 \le i < j \le 16$.