To solve the problem, we need to find the minimum value of the expression \(|MA - NB|\), where:

- Points \(A(-2, -2)\) and \(B(6, 8)\) are defined.
- Point \(M\) lies on the line \(Ox\) (the x-axis), which is where \(y = 0\).
- Point \(N\) lies on the line segment \(MN\) where \(M = (x, 0)\) and \(N = (0, 1)\).

### Step 1: Define Points
Let:
- \(M = (x, 0)\)
- \(N = (0, 1)\)

### Step 2: Calculate Distances
The distances from \(M\) to \(A\) and from \(N\) to \(B\) are given by:

1. Distance \(MA\):
\[
MA = \sqrt{(x - (-2))^2 + (0 - (-2))^2} = \sqrt{(x + 2)^2 + 2^2} = \sqrt{(x + 2)^2 + 4}
\]

2. Distance \(NB\):
\[
NB = \sqrt{(0 - 6)^2 + (1 - 8)^2} = \sqrt{(-6)^2 + (-7)^2} = \sqrt{36 + 49} = \sqrt{85}
\]

### Step 3: Set Up the Expression
We need to minimize:
\[
|MA - NB| = \left| \sqrt{(x + 2)^2 + 4} - \sqrt{85} \right|
\]

### Step 4: Analyze the Expression
Setting:
\[
\sqrt{(x + 2)^2 + 4} = k
\]
We need to find \(k\) such that:
\[
k \rightarrow \sqrt{85}
\]

### Step 5: Solve for Critical Points
The square root function is increasing. Therefore, to minimize \(|MA - NB|\), we need to find values of \(x\) such that:
\[
\sqrt{(x + 2)^2 + 4} = \sqrt{85}
\]

Squaring both sides:
\[
(x + 2)^2 + 4 = 85
\]

This simplifies to:
\[
(x + 2)^2 = 81
\]

Taking the square root yields:
\[
x + 2 = \pm 9
\]

So:
1. \(x + 2 = 9 \rightarrow x = 7\)
2. \(x + 2 = -9 \rightarrow x = -11\)

### Step 6: Evaluate at Critical Points
- For \(x = 7\):
\[
MA = \sqrt{(9)^2 + 4} \equiv \sqrt{81 + 4} = \sqrt{85}
\]
\(|MA - NB| = |0| = 0\).

- For \(x = -11\):
\[
MA = \sqrt{(-9)^2 + 4} \equiv \sqrt{81 + 4} = \sqrt{85}
\]
\(|MA - NB| = |0| = 0\).

### Conclusion
Thus, the minimum value of \(|MA - NB|\) is:
\[
\boxed{0}
\]