Sounds like the belt is squealing whenever your power steering pump is under load. I had a similar issue with my '03 Rio, whenever I turned on anything electric the extra load on the alternator made the belt squeak AND it also squealed when I started the car up and the engine and belt were cold. In my case the alternator bracket was broken and wasn't allowing the belt to tighten enough (my dealer broke it and tried to hide it...)
Unfortunately I don't know much about the '06 and up Rios, I don't know what your belt configuration is but I'd start by seeing if the belt for your p/s pump is loose or not, it's possible they used the wrong sized belt (being lazy) and the adjuster is maxed out. You could try spraying some belt treatment on your pulleys to clean them and dry them out, it's possible they spilled something on the pulleys when they did your timing belt and it's slipping.
My cars: 2003 Kia Rio, 1977 Dodge W300 Wrecker, 1986 Chevy Monte Carlo, 1977 Ford F250 Wrecker, 1985 Subaru GL, 1986 Audi 5000S, 1990 E150, 1983 Chevy Monte Carlo, 1982 Ford Mustang, 1978 Ford F150, 1988 Mercury Grand Marquis