Find the greatest common divisor (GCD) of the lengths of the timber pieces to determine the maximum possible equal piece length.

The lengths are 240 cm, 320 cm, and 380 cm.
Prime factorization of each length:

$$240 = 24 \times 10 = (2^3 \times 3) \times (2 \times 5) = 2^4 \times 3 \times 5$$ $$320 = 32 \times 10 = (2^5) \times (2 \times 5) = 2^6 \times 5$$ $$380 = 38 \times 10 = (2 \times 19) \times (2 \times 5) = 2^2 \times 5 \times 19$$

The common prime factors are 2 and 5.
The lowest power of 2 is $2^2$.
The lowest power of 5 is $5^1$.

$$\text{GCD} = 2^2 \times 5^1 = 4 \times 5 = 20 \text{ cm}$$

The maximum possible equal piece length is 20 cm.

The side length of the square is 20 cm.

$$\text{Area of square} = \text{side} \times \text{side}$$ $$\text{Area} = 20 \text{ cm} \times 20 \text{ cm} = 400 \text{ cm}^2$$

The highest possible area of a square that can be made from any of the 3 pieces is $400 \text{ cm}^2$.