That question is too abstract for me to reason about. A "rectangle" couldn't move at the speed of light; light is point-like (oh dear, saying that could get me in trouble). Anything with mass cannot move at the speed of light, and anything without mass must move at the speed of light, and all known objects without mass are point-like. And then there's the question of observation in the first place - observation itself is governed by fundamental particles, and observing a fundamental particle at its scale is theoretically impossible.

I know you want me to cheat and set aside all of these complications and just imagine a rectangle moving at the speed of light, but such an answer would be nonsensical.

But the answer to the more specific question, "does the expansion of space make something move faster than the speed of light?", is "no", but it does cause the distance between objects to increase faster than the speed of light. To an external observer (which is inherently difficult to reason about, for the record), these objects would appear to "move" away from one another faster than lightspeed. However, the very idea of "movement" is rooted in the context of spacetime, and is bounded by the speed of light, so it's a confusing/nonsensical question.

See also the observable universe: https://en.wikipedia.org/wiki/Observable_universe. As the universe's expansion speeds up, we will never receive further information from some parts of the universe as it "moves away from us" faster than the speed of light.