Let $m > 1$ be an integer.  A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$, $a_3 = 4$, and for all $n \ge 4$, $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$

Determine all integers $m$ such that every term of the sequence is a square.
1. **Initial Terms and Recurrence Relation:**
   Given the sequence \(a_1, a_2, a_3, \ldots\) defined by:
   \[
   a_1 = 1, \quad a_2 = 1, \quad a_3 = 4, \quad \text{and} \quad a_n = m(a_{n-1} + a_{n-2}) - a_{n-3} \quad \text{for} \quad n \geq 4
   \]
   We need to determine all integers \(m\) such that every term of the sequence is a perfect square.

2. **Calculating Initial Terms:**
   Let's compute the first few terms using the given recurrence relation:
   \[
   a_4 = m(a_3 + a_2) - a_1 = m(4 + 1) - 1 = 5m - 1
   \]
   \[
   a_5 = m(a_4 + a_3) - a_2 = m((5m - 1) + 4) - 1 = m(5m + 3) - 1 = 5m^2 + 3m - 1
   \]
   \[
   a_6 = m(a_5 + a_4) - a_3 = m((5m^2 + 3m - 1) + (5m - 1)) - 4 = m(5m^2 + 8m - 2) - 4 = 5m^3 + 8m^2 - 2m - 4
   \]

3. **Condition for Perfect Squares:**
   We need \(a_4, a_5, a_6, \ldots\) to be perfect squares. Consider the product \(b = 4 \cdot a_4 \cdot a_6\):
   \[
   b = 4 \cdot (5m - 1) \cdot (5m^3 + 8m^2 - 2m - 4)
   \]
   Simplifying \(b\):
   \[
   b = 4 \cdot (25m^4 + 40m^3 - 10m^2 - 20m - 5m^3 - 8m^2 + 2m + 4) = 4 \cdot (25m^4 + 35m^3 - 18m^2 - 18m + 4)
   \]
   \[
   b = 100m^4 + 140m^3 - 72m^2 - 72m + 16
   \]

4. **Comparing with Perfect Squares:**
   We compare \(b\) with the squares of polynomials:
   \[
   (10m^2 + 7m - 7)^2 = 100m^4 + 140m^3 - 91m^2 - 98m + 49
   \]
   \[
   (10m^2 + 7m - 6)^2 = 100m^4 + 140m^3 - 71m^2 - 84m + 36
   \]
   We need:
   \[
   b \leq (10m^2 + 7m - 7)^2 \quad \text{or} \quad b \geq (10m^2 + 7m - 6)^2
   \]

5. **Case Analysis:**
   - **Case 1:** \(b \leq (10m^2 + 7m - 7)^2\)
     \[
     100m^4 + 140m^3 - 72m^2 - 72m + 16 \leq 100m^4 + 140m^3 - 91m^2 - 98m + 49
     \]
     \[
     -72m^2 - 72m + 16 \leq -91m^2 - 98m + 49
     \]
     \[
     19m^2 + 26m - 33 \leq 0
     \]
     This inequality is not true for any positive integer \(m\).

   - **Case 2:** \(b \geq (10m^2 + 7m - 6)^2\)
     \[
     100m^4 + 140m^3 - 72m^2 - 72m + 16 \geq 100m^4 + 140m^3 - 71m^2 - 84m + 36
     \]
     \[
     -72m^2 - 72m + 16 \geq -71m^2 - 84m + 36
     \]
     \[
     m^2 - 12m + 20 \leq 0
     \]
     This inequality holds for \(2 \leq m \leq 10\).

6. **Checking Specific Values:**
   - For \(m = 2\):
     \[
     a_4 = 5 \cdot 2 - 1 = 9 \quad (\text{perfect square})
     \]
     \[
     a_5 = 5 \cdot 2^2 + 3 \cdot 2 - 1 = 31 \quad (\text{not a perfect square})
     \]
   - For \(m = 10\):
     \[
     a_4 = 5 \cdot 10 - 1 = 49 \quad (\text{perfect square})
     \]
     \[
     a_5 = 5 \cdot 10^2 + 3 \cdot 10 - 1 = 529 \quad (\text{perfect square})
     \]
     \[
     a_6 = 5 \cdot 10^3 + 8 \cdot 10^2 - 2 \cdot 10 - 4 = 7744 \quad (\text{perfect square})
     \]

7. **Conclusion:**
   The only values of \(m\) that make every term of the sequence a perfect square are \(m = 2\) and \(m = 10\).

The final answer is \( \boxed{ m = 2 } \) and \(m = 10\).