No, polarized caps can be replaced by non-polarized (though on googling, there seems to be a lot more confusion on the topic than I expected--and it's true that you generally can't go the other way around, replacing non-polarized with polarized).

The only reason polarized caps are ever used is because they're cheaper and smaller per uF than other technologies.  But recently, MLC Caps have come down so much in price that they make a viable replacement even for caps into the 47 uF range.  There's some good commentary in this thread .

I'm going to give it a more thorough washing, using soap this time, and try again to trace C4.  While it used to be checkerboard, now I get vertical stripes occasionally on boot.