Those squares are the borders between two words that are two different colors. Since each color represents a different continent, that effectively makes them the borders between continents.
For example, 10-Across is part of the Red continent, and it's five letters long. Its last letter is also the middle letter of 8-Down, part of the Purple continent. Because the two words cross, 10-Across and 8-Down border on each other. A player who controls the Red continent can make inroads into Purple by attacking 8-Down from 10-Across.
Just like in a crossword puzzle, 1-Across and 1-Down are two different words (and thus, two different countries). Since they cross each other (in the upper-left square of the grid), they also share a border.