Actually, maybe this has changed recently but euros definitely do NOT go everywhere in the country. I watched one old grandma try to pay in the grocery store, and trust me the reaction was amusing to say the least
. Maybe since their recent vote on foreigners....
BUT, with that said, these huts are on the border and I am quite sure they do take both. If it is a swiss hut, they prefer CHF, and may ONLY give you change back in CHF even if you pay in euros, but if it is an italian hut it is probably the reverse. Some also may take visa now which is awesome, you can check their websites. I also highly recommend a reservation.
http://www.sac-cas.ch/en/huts.html, I am sure you probably already have it, but there is all the info you will need. For example you can make reservations for the Monte Rosa hut online.
For coverage, from the Swiss side of the border, I can recommend STRONGLY swisscom. They are a bit more expensive, although for a prepaid phone card its like only a few francs, but they have hands down, the best, back mountain coverage. I personally have an Orange company Iphone, but has a small handy to take in the mountains that is Swisscom. The reception may not be perfect, but from the huts there usually is some signal, and actually all the summits we have been on as well. If you are spending more time on the Italian side of things, I couldn't say but the coverage up high should be the same. To simply buy a sim card which is prepaid, I cannot imagine them asking to see a passport. In the huts you also should not need it as well, unless there is some sort of emergency and they need the information there, but I cannot picture that being the case.
Have fun!