In my experience, not really. The round male pins are too thin for square female, and male square are too big for round female.
I use square for almost everything. (Also because round don't seem to be available for cheap on eBay in right angle versions).
However, the round female are extremely good as component sockets, for example if you want to easily swap LEDs, capacitors, etc. on a PCB. And the round male pins work reasonably well, soldered to thin wire, to insert into IC sockets without damaging them.
Beware, if you insert a too large pin (regardless of shape) into a round female socket, it will ruin the contacts inside and leave it loose. [Insert juvenile joke about loose female holes here.]