1. Understanding the Problem:
We need to find all positive integers $n$ less than 300 such that $n$ has one less positive divisor than $n + 1$. This means:
$$d(n) = d(n+1) - 1$$
where $d(x)$ denotes the number of positive divisors of $x$.

2. Analyzing the Divisors:
For $d(n)$ to be one less than $d(n+1)$, either $n$ or $n+1$ must be a perfect square. This is because a perfect square has an odd number of divisors, while a non-perfect square has an even number of divisors.

3. **Case 1: $n$ is a perfect square:**
Let $n = k^2$. Then $n+1 = k^2 + 1$.
- $d(n) = d(k^2) = 3$ (since $k^2$ has an odd number of divisors).
- $d(n+1)$ must be 4 (since $d(n+1) = d(n) + 1$).

4. **Case 2: $n+1$ is a perfect square:**
Let $n+1 = m^2$. Then $n = m^2 - 1$.
- $d(n+1) = d(m^2) = 3$ (since $m^2$ has an odd number of divisors).
- $d(n)$ must be 2 (since $d(n) = d(n+1) - 1$).

5. Finding Stepstool Numbers:
We need to find all $n$ such that either $n$ or $n+1$ is a perfect square and $n < 300$.

- For $n = k^2$:
$$k^2 < 300 \implies k < \sqrt{300} \approx 17.32 \implies k \leq 17$$
So, $k$ ranges from 1 to 17. The corresponding $n$ values are:
$$1^2, 2^2, 3^2, \ldots, 17^2 = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289$$

- For $n+1 = m^2$:
$$m^2 - 1 < 300 \implies m^2 < 301 \implies m < \sqrt{301} \approx 17.37 \implies m \leq 17$$
So, $m$ ranges from 1 to 17. The corresponding $n$ values are:
$$1^2 - 1, 2^2 - 1, 3^2 - 1, \ldots, 17^2 - 1 = 0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288$$

6. Combining and Summing Unique Stepstool Numbers:
Combining the two sets of $n$ values and removing duplicates, we get:
$$\{0, 1, 3, 4, 8, 9, 15, 16, 24, 25, 35, 36, 48, 49, 63, 64, 80, 81, 99, 100, 120, 121, 143, 144, 168, 169, 195, 196, 224, 225, 255, 256, 288, 289\}$$
Summing these values:
$$0 + 1 + 3 + 4 + 8 + 9 + 15 + 16 + 24 + 25 + 35 + 36 + 48 + 49 + 63 + 64 + 80 + 81 + 99 + 100 + 120 + 121 + 143 + 144 + 168 + 169 + 195 + 196 + 224 + 225 + 255 + 256 + 288 + 289 = 687$$

The final answer is $\boxed{687}$.