Popsicle stick, sure why not. As good as another easy to handle item that's about the same thickness.
However there are other ways to try quieting a brake squeal. Toe out, grinding small ramps to the pad's rear facing edges, wiping garden dirt on the pads (my favorite partially because so many cringe at the thought of making a brake dirty as a good thing
) and trying different pads.
BTW can you provide an image of these Phil brakes? Having followed Phil Wood products for decades this item is a new one to me. Perhaps you mean "Paul" brand? Andy