The problem gives us two quadratic equations with their roots being mutually defined:

1. The roots of the equation x^2 + mx - n = 0 are p and q.
2. The roots of the equation x^2 + px - q = 0 are m and n.

Let's analyze the statements:

Statement I: m(m+n) = -1

Using Vieta's formulas for the first equation x^2 + mx - n = 0:

• p + q = -m
• pq = -n

From the second equation x^2 + px - q = 0 using Vieta's formulas again:

• m + n = -p
• mn = -q

We have the following expressions:

• p + q = -m
• pq = -n
• m + n = -p
• mn = -q

Rewriting the expression for m(m+n):

• m(m+n) = m(-p) = -mp

We also know from the roots of the equation that:

• m = \frac{q}{n}
• n = pq
• Combining with mn = -q, we have m = \frac{q}{n} and n = -\frac{q}{m}
• Thus, m(m+n) = m \left( - \frac{q}{m} \right) = -q and from the initial mn = -q, it implies that m(m+n) = mn.
• If m(m+n) = -1, we then conclude that mn = -1.

This confirms that Statement I: m(m+n) = -1 is correct.

Statement II: p + q = 1

We know from Vieta's formulas that p + q = -m. Without additional constraints relating m to 1, we cannot conclusively say that p + q = 1. Therefore, this statement is not necessarily true given the available information.

Hence, only Statement I is correct.

Conclusion:

Therefore, the correct answer is I only.