Illustrator doesn't have a table facility (that I know of), so you might have to make it manually. There are a few ways of doing it, here's one:
1: Draw a rectangle the size you need for the overall crossword, with a white fill and black stroke.
2: Go to Object > Path > Split into Grid
3: In the options dialog, change both gutters to 0, and enter the number of boxes you need horizontally and vertically.
4: Click OK, then select any squares you want to be black and colour them appropriately.