If the nails held in the holes you drilled for tapcons, either...
1. you had *huge* nails

I used nails my carpenter friend gave to me. He called them masonry
nails or something like that. They are big and ribbed. Trying to nail
them in the old concrete just resulted in sparks and lots of concrete
breaking up. The drilled hole in the concrete was just a little
smaller than the nails, so there was enough pressure to hold them once
nailed in.

2. the holes were the wrong size for screws...might be why they broke.

I bought a pack of screws that came with a masonry bit, so unless
there was a packaging error it should be the right size. The screws
started going in but about halfway through, they came to a dead stop
even though the hole was deep enough. There seems to be some small
rocks in my concrete, rocks that are hard as hell to drill through, I
think the screw hit one of them.

As I said I'm no pro so I normally go by trial and error until I find
something that is good enough.

Max.