You are confusing a couple terms. The machine they are sewing the pleats with in the video is not a true walking foot machine. It is compound feed, however. Compound feed means more than one type of feed system working together, usually drop feed (using feed dogs) and needle feed. On a compound feed machine, the needle rocks back and forth to help move the work through the machine. Walking foot is a generic term for alternating presser feet. A walking foot machine's foot is in two parts, the outside section is attached to a separate shaft, as is the inside section of the foot. When one part of the foot is up, the other part of the foot is down, and vice versa. Hence the name "walking" foot. The intention of this is to always have some part of the foot down to hold the work in place better. The needle is also mounted to it's own separate shaft. If the machine does not have three separate shafts, it can't have alternating presser feet, and is therefore not a "walking foot" machine. There is no attachment that can turn a non walking foot machine into a true walking foot machine, or change it to a compound feed machine.
Your squares are just straight lines in two different directions, it's the same thing.
Buy sew foam with a fabric backing, not scrim. Scrim is not near as durable, and won't hold up to being sat on as well. It is cheaper, though.
BTW, the way they are sewing the pleated top to the seat is backward. I sew the pleats to the foam from the top of the fabric so I can see what is happening on the good side, not the side nobody will ever see. On anything more elaborate than what they are sewing, you could really end up with a mess, especially 1" squares. No way they will look good if sewn from the back side, especially sewing to 1/2" thick sew foam like they are. Your home machine will not handle 1/2" sew foam, so don't even try. 1/4" foam is all it will handle, and maybe not that.
Try sewing the design that's on this bike seat from the back side and see what happens.
Please define "pleather" for me.