The bales of hay lying around in the fields here are the round ones.
Do they even bale hay in squares anymore?
I haven't seen those in years.
Yes, the square bales are still relatively common.
There's several factors on which baler type is used. Here's some common ones.
Terrain - If you're in hilly terrain, round bales can be a problem.
Keeping them from rolling down the hills as well as moving them without tipping.
Size and age of farm - Smaller and older farms still do a lot of the square bales. The older barns have the hay loft so you put the squares in there. Newer farms don't have lofts. Also, a small farm may have a hard time justifying the cost of a round baler (next point).
Cost - Balers aren't cheap. An old square baler (20-40 years old) in good, operating condition will still set you back a few grand. Round balers can easily fetch $10k+.
As for how you collect your square bales, there's 3 common ways.
Kick baler - The baler has a kicker in the back that throws the bale up in the air. When you have a wagon (with sides) in tow behind the baler, the bales are shot directly into the wagon.
Chute into wagon - Again, with a wagon in tow, there's an extension on the back of the baler that allows someone standing/riding in the wagon to grab the bale, move it to the back and stack.
Walk it. You bale with the bales landing on the ground. When done baling, you disconnect the baler and connect a wagon. One driver going down the field slowly while a few people pick up the bales and get them on the trailer. Time and resource consuming but it gets it done.