I'm thinking there's probably no reason to devote time to words and photos explaining how I cheated the yarns into place. They only needed to be moved from a connected corner either right to the next corner or up to the next corner. Moving right was super easy and I got in the habit of pulling the last yo back out of the last dc made and completing that yo with the new color. It drapes a short strand of yarn across the chain 3 space that you'll be crocheting into. So the 3 dc that go into that chain 3 space will cover it up nicely.
Moving the yarn up: first I tried catching the strand in the last dc on the left of the square before (moving diagonally; the rows march diagonally, up and left each block to make a row). Just put the strand over or under the working yarn before pulling a yo through to make the dc. (This does leave little contrasting dashes, but they're next to the color they belong with.) Then I finally got the bright idea to borrow from the original pattern. I started using slip stitches. When the color needed to move up to the next corner it would have been going from the lower right corner of a square in the previous row to the upper right corner of the same square. The loose strand was just waiting there in the lower corner. So I started making a loop through the top of the lowest stitch in the square and then making slip stitches up the top of the stitches next to it.
And then by the time I got to here I realized that many words about crochet without pictures to back it up is a disaster. Fortunately I took pictures while cheating the colors into place. Uncertain as to the quality, but here goes! Well, backing up a minute. Overall I'm making something like the blue diagonals square, which is four 6"x6" squares sewn together, but crocheting the 12"x12" in one piece. I'm using SmoothFox's Diagonal Box Stitch Square 6x6. It's not designed to be use this way, but I was determined to tweak it into what I wanted. When the diagonal stripes need to go in a different direction the color strands needed are not quite where they need to be, but close.
Leave that loop on the hook for now. Draw a loop through the top of the next stitch and pull it through the first loop. That's making slip stitches to bring the yarn up to where it needs to be.
Next the blue loop is going to need to be where the yellow white one is. It's replacing the last yo and draw through in the last dc in the yellow white block. That follows the theory that when you want to change colors you change in the last yo of the stitch before (or else the next stitch you make is wearing a hat that doesn't match).
So carefully pull the yellow/white loop out, put the two loops it held on the hook, put the hook through the blue loop
I'm not the best explainer, but these notes will help me to recreate it the next time I work this pattern this way. It's written to just have the colors going in one diagonal direction. Doing that doesn't require the threads to be moved to the next open corner. I just wanted the effect of having sewn 4 small squares together without having to do it. The construction of the diagonal box pattern doesn't really leave good edge stitches or any selvedge for nicely sewing through. Then there's the reality that I don't know crocheting nearly as well as knitting. Could be this is very very basic crochet technique that I just never learned.