Consider a $9 \times 7$ rectangle that needs to be tiled using two types of tiles: L-shaped tiles, which cover three unit squares, and $2 \times 2$ square tiles, which cover four unit squares. We need to find all possible values of $n$, the number of $2 \times 2$ square tiles used in such a tiling configuration.

### Step 1: Calculate Total Area

The area of the rectangle is:
$$9 \times 7 = 63 \text{ unit squares}.$$

### Step 2: Analyze Tile Coverage

- Each L-shaped tile covers 3 unit squares.
- Each $2 \times 2$ square tile covers 4 unit squares.

Let $x$ be the number of L-shaped tiles used, and $n$ the number of $2 \times 2$ square tiles used. Thus, we have the equation accounting for all unit squares covered:
$$3x + 4n = 63.$$

### Step 3: Determine Possible Values for $n$

The equation $3x + 4n = 63$ must be satisfied with non-negative integer solutions.

First, let's solve $3x + 4n = 63$ for valid $n$:

1. **For $n = 0$:**
$$3x = 63$$
$$x = 21$$
Therefore, when $n = 0$, $x = 21$, which is a valid tiling configuration.

2. **For $n = 1$:**
$$3x + 4 = 63$$
$$3x = 59$$
This does not yield an integer value for $x$.

3. **For $n = 2$:**
$$3x + 8 = 63$$
$$3x = 55$$
Again, this does not yield an integer value for $x$.

4. **For $n = 3$:**
$$3x + 12 = 63$$
$$3x = 51$$
$$x = 17$$
Therefore, when $n = 3$, $x = 17$, which is a valid configuration.

5. **For $n \geq 4$:**
The equation $3x + 4n = 63$ quickly becomes impossible to satisfy with integer values since each increase by 1 in $n$ requires 4 more unit squares, but the remaining must still be divisible by 3.

Thus, the values of $n$ that allow for a perfect tiling of the rectangle are:
$$\boxed{0 \text{ and } 3}.$$