Do they have highway rest stops (restrooms) in Switzerland and Austria?

Yes, they have rest stops. They usually charge to use the restrooms, but your little receipt often gives you that amount off a purchase in the store/coffee shop.

Be sure to get an Austrian vignette as you enter the country (good potty stop opportunity). If you are renting the car in Switzerland, it should already have a Swiss vignette attached.

Rest stops in Austria and Switzerland differ from rest stops in US. There are not only restrooms there but usually also restaurant, cafeteria, gas station, sometimes repair shop, motel.

thank you for the replies. What do they call them and are there signs for them one can see while driving?

They are not very frequent, so in case of need stop at the first you meet as the following one may be after one hour or more driving.

There are not only restrooms there but usually also restaurant,
cafeteria, gas station, sometimes repair shop, motel.

Rest stops on interstates in the Northeast of the USA have these services as well, minus the motel.

for switzerland https://www.autobahnen.ch/dev/index.php?lg=001&page=018 or http://acs.ch/ch-fr/pdf-public/Reisen/Autobahn-Restaurants_und_Tankstellen_in_der_Schweiz.pdf

for austria http://www.asfinag.at/road-safety/resting/rest-areas/ also see Toll/ Toll Sticker section regarding need for a vignette on most Austrian autobahn

They're called "Rastätte" or "Autobahnrastätte" (because they are on Autobahns) in German, but for the first part of your trip from Geneva you will be in French speaking Switzerland so they will be called "Aire de repos".

Here is a website with a map and list of all of them in Switzerland, plus links to their individual websites: https://www.autobahnen.ch/dev/index.php?lg=001&page=018

Here is a leaflet listing all the ones in Austria, with facilities: http://www.arboe.at/fileadmin/uploads/ARBOE/kat/reise/tanken_rasten_2012.pdf

They are signposted symbolically, with a knife and fork and petrol pump. For example this sign is advance warning of one in 6 Km (and the next fuel after that is 92 Km): https://upload.wikimedia.org/wikipedia/commons/c/c9/Hinweisschild_Autobahnrastst%C3%A4tte_Altenburger_Land.JPG

Note a Tankstelle is fuel only, a Rastätte includes food as well. Both will have toilets.

Thank you so much!

In my opinion, the best rest stops I've ever visited were in Switzerland :-). The ones we stopped at were like luxury stops compared to the ones I'm used to in my area. Fresh flowers, gourmet food stations, caviar anyone? And of course Swiss chocolate :). I'd say those stops were definitely a big plus to our travels!!