Is there any way to force square header pins into round holes?
I don't know how hard you are pushing them, but if you put the header on a table and push the board down onto it you can get enough pressure to potentially damage the board. 

Sounds like they made the holes a bit on the small size. Most pins for 0.1" pitch headers are 0.025" square with slightly rounded corners. You may or may not find that an alternate supplier has ones that fit a bit better, since you're down to the tolerance.

Usually the header manufacturers recommend 1mm (55.7 mils) vs. 35 mils for the pin diagonal. I think 0.9mm is a bit better for hand assembly- it's theoretically size-on but the rounding of the corners makes it a snug fit usually.