Foam comes in different types. The most common foam to use for armor is EVA foam. This comes in many forms, from the anti-fatigue floor mats stated above, to the "Foamies" you find in craft stores. It's all the same stuff, just different thicknesses. It easily forms with heat, and you can obtain rounded objects by forming the foam over bowls, styrofoam balls, etc. The really squishy stuff you're referring to might be upholstery foam (usually green, found in fabric stores) used for cushions, pillows, etc. This foam, while not necessarily good for stiffened-but-flexible armor, is useful for padding and fabric applications (muscle suits, sewn-in details, etc).
Wonderflex is a form of thermoplastic. It comes in sheets, and it becomes pliable and formable when warmed with a heat source (blow dryer, heat gun, over the stove, etc). Once cool, it holds the shape and becomes hard. Like foam, you can form it over round objects for things like shoulder pauldrons or breastplates.
Sandwiching craft foam (which is just a thin version of EVA foam, like the floor mats) between layers of wonderflex is common practice. You'll still need to coat the wonderflex, as it has this bumpy, waffley grid texture in its normal state, but it can provide lightweight, easy to form armor.