So I am making a game like this: http://i.stack.imgur.com/89zc4.jpg

and not sure if to use hex or square

what would be better

Squares are the most common. They are the easiest to lay down, its easy visually track distance, moving in straight lines is possible.

The biggest downfall to them is diagonals. Say each square is an inch. If you move one square in any cardinal direction, you’ve moved one inch. If you move diagonally to the NE, you are moving 1" up, and 1" left, so using pythagorean theorem, you would take the square root of 1+1 (since 1 squared is still 1), SQRT(2)= 1.41, so if you move diagonally you’re actually moving 1.4 inches.

Which is why in D&D (where each 1 inch square is considered 5 feet), when moving diagonally, the first square is 5 feet, the second is 10, the third is 5 and so on. So if you move 4 squares diagonally, you didnt move 20 feet, you moved 30.

That is the biggest advantage (that I know of) to Hex grids. No matter which of the 6 directions you go, if your hexagons are 1" moving from the center of one to the center of any of the 6 surrounding hexagons is only 1"

Thats why highy tactical games tend to use hex grids.