Illustrator fill question- center one design in a shape

As long as your patterns are the same size as your grid squares, and your grid originates at 0,0 x/y position on the canvas, you can use pattern fills and live paint:

Here's an older post about knitting patterns, for reference:

https://community.adobe.com/t5/illustrator/centering-text-in-grid-box/td-p/10316152?page=2

Oh, and you can change the color of individual symbols (so your X stitches) by selecting them and adding a fill in the Appearance panel.

I would recommend using Symbols, like Ton said.

Make your X, go to the Symbol panel, and either by going to the panel menu or clicking the 2nd button at the bottom of the Symbol panel, make a new Symbol.

Make a rectangle the same size as that X, and make it a symbol, too. Use the box symbol to make your cross-stitch canvas. You can Alt (PC) /Option (macOS) + drag to make one copy next to that box. Use Ctrl (PC)/Command (macOS) + D to duplicate the box that same distance and direction however many times you want. Then select the row and do the same to duplicate the rows.

Then you can Shift + select the boxes where you want a stitch, go to the Properties panel to the lower part of the panel where it says Replace Symbol. Select the X from the dropdown menu, and Illustrator will swap it with all the box symbols you have selected. 

You made a pattern out of the red X, that is difficult to control because it repeats.

You could drag the red X to the symbols panel and drag it from the symbolls panel to the artboard when you need it.

Edit: I would make it a Static Symbol, which makes it easier to align with Smart Guides on.