You still don't need the driver disk, it obviously recognizes the NIC, the problem you have is probably one of auto negotiation which is a SWITCH issue not a NIC problem.
Try forcing the SWITCH (port where ESX is plugged into) to 10/100 FULL DUPLEX, and go from there.
We have many machines with BROADCOM Netextreme's and not 1 problems with connectivity, so that's the problem your SWITCH. What happens when you install Windows on those SAME ESX host to that SAME switch, same issue?