Eastern stitch is also known as Egyptian buttonhole. I guess the buttonhole part of the name comes from the loop made between the two bars as the same hand movement is required to make buttonhole. I have seen it classified as part of the buttonhole family. Usually this stitch is seen worked as a filling particularly in canvas work embroidery.
This stitch works with any grid like pattern. I have seen Eastern stitch as it is worked row upon row but first pointed in one direction and then on the next row point the stitch in the other direction it works because as a stitch each unit will sit neatly within a square. The beaded version is the same for the same reason.
Since it sits so firmly with in a square any grid like pattern can be created by treating it like cross stitch and simply counting the squares. You can also spin it around to create patterns. It is just a case of looking at each unit and thinking slightly differently.
Many people think of this stitch as an evenweave stitch that has to be worked on a grid but it will follow a softly curved line too.
How to work Eastern Stitch
Work this stitch from left to right. When working this stitch it needs to kept at a fairly loose tension.
Make a vertical straight stitch at the left side of an imaginary square.
Make a horizontal straight stitch at the top of an imaginary square.
These two stitches are the foundation bars of each unit of eastern stitch.
Taking the needle diagonally across the back of the fabric, bring the thread up to the front the fabric, in the bottom right-hand corner as illustrated.
Take the needle over the top and then slide it under the vertical stitch. When you slide the needle under the bar make sure it points towards the bottom of the square. Pull the needle through so that it forms a loop over the vertical bar stitch as illustrated. You are not going through the fabric but lacing the bar.
Move to the horizontal bar and slide the needle under the horizontal stitch.
Make sure the thread is kept to the right of the needle so that when you pull it through it forms a loop over the horizontal stitch as illustrated
Take the thread through the fabric at the bottom right-hand corner.
Bringing the needle out at the top left corner of the next stitch and repeat the process along a line or to fill an area.
You can work along a line or a curve or as individual units to create patterns.