[ZenGum]

Okay, the stuff about pumpkin seeds is distraction.

I want to start by understanding the question. Here are some clear points for starters:

1. We take blocks of a regular size, 1x1x1.
2. We arrange some into a perfect cube.
3. We arrange others into a perfect square. *see post 105
4. The number of blocks used to make the cube must be equal to the number of blocks used to make the square.
5. Adding these numbers together is the final answer.
6. The cube must be able to fit inside the square.

Now for some less certain parts of the question:
7. There is exactly one correct answer.
8. This must be an integer.

Now for some assumptions which are not stipulated. The solution probably lies in challenging one or more of these:
9. The square must touch the cube at at least one point.
10. The square must abut the cube along all of the cube's faces.
11. The square is only one layer high.
12. The square is only one layer thick.
13. The square is on the same alignment as the cube.

Violating 10 makes it fairly easy. You could have a cube 4x4x4, having 64 blocks, and make a square 16 blocks each side around it, thus using 16x4 = 64 blocks, thus reaching HLJ's answer of 128. Or you could make the square double height, 8 blocks per side, and still use 64 blocks. *see post 105

This does not deliver a uniquely correct answer, since other combinatiosn work with this.
You could have a cube of 6x6x6 = 216 blocks, and a square 8 blocks per side and 6 blocks high (8x4)x6 =216 blocks.

Or a cube 100x100x100 = 1,000,000 and a square 250 long and 1,000 high (250x4)x1000 = 1,000,000.

If you reject 10 and 13 you can play around with Pythagorean triads, but I had a look and couldn't find anything promising.

What the hell kind of pumpkin has cubic seeds, anyway?