What type of screw terminals are better for connecting solid core wires to PCB, the square hole or round hole type?
Have you read the specification sheets?
I mean is one mechanism type generally considered higher quality?
Used both, Not seen any difference.
For what kind of wire? That is why I pointed you to the specs. Probably, each type is optimized for a certain range of wire types.
The flat type would probably accept a wider range of wire guage.
"quality" doesn't exist outside of specifications.
Logically, the round hole type will tend to centre the solid core wire as it is tightened, but on the other hand, since both types you show have a pressure leaf to prevent the screw mashing the wire, that actually prevents complete closure on the round hole type and it will fail to hold too small a wire, especially if stranded. 
Suggest either, also double up the stripped end of the wire for 24 and 26 AWG.
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