How can I remove the bolts holding down my utility sink?
Those look like concrete nails to me. Might have been hand-hammered in, but if it's somewhat recent, it was probably driven in with a powder-actuated system. 

You don't usually try to pull these out. My advice is to cut/grind away the head. You can do this with a Dremel tool and an abrasive disk; it will take a while, but it's simple to do and easy to control.

Or, you can use an angle grinder. That will take away the head much more quickly, but it's a little harder to control. Make sure to use the screw-in handled if you do this. 

Once the head it gone, remove the sink, and then either cut off the protruding fastener or just grind it away.