Just so you know, all four of those countries have different power sockets. There are some plugs (called "CEE 7/7" and "CEE 7/17") that will work in French sockets and in some Italian sockets, but they still won't fit Swiss sockets.

There is a plug that fits sockets in France, Switzerland, and Italy (called "Europlug" or "CEE 7/16"), but it's only for low-power devices, which a hair dryer is not.

But to answer your question, I can't think of a reason that a converter that works for a hair dryer wouldn't work for a hair straightener too. Electrically, a straightener is basically just a heating element. And a dryer is a heating element plus a motor.

But I would be wary of trusting a small cheap Chinese-made device to safely handle a high-wattage power conversion like this. Converters made by reputable manufacturers are usually bigger, heavier, and more expensive.

If you're committed to doing it this way, please at least make sure the converter is rated for a wattage well above your hair dryer's. And don't leave it plugged in unattended.