I agree the .025 square pin headers will make the breadboard clips loose and problematic, especially for round wire component leads so I avoid them for breadboarding purposes.
I use the cheap, round pin headers available from China, gold plated, .017 diameter pins. About 2 cents/pin in lots of ten strips of 40 pins. eBay and Aliexpress have plenty of listings. I don’t know the pin length, can measure if it matters. BTW, the longer, thin ends go into the BB.
I concur: get cheap Chinese machined pins. They work great.
I've seen a hole or two go bad on a breadboard, but that's what they do. They should be thought of as disposable.
A high quality breadboard will last for many, many, many years without getting loose, if it's treated properly (i.e. adhering to the maximum wire size). Cheap ones are a different matter…
Square pins really are too big, and will wear out a cheap breadboard quickly. Even worse are things like many potentiometers, TO-220 semiconductors, etc. The latter can be made safe by using flat jaw pliers to rotate the pins by 90 degrees, so that the pin enters parallel to the terminal strip, instead of prying it apart.