Strange, the vmkernel supposedly detected and claimed the devices, assigning them to vmnic2 and vmnic3 (see the "VMkernel Name" in the detailed hardware output), which means the driver should be ok.

Can you run these commands?:

# ethtool -i vmnic2

# esxcli network nic get --nic-name vmnic2

# grep -i vmnic /etc/vmware/esx.conf

# esxcli system module get --module igb

# esxcfg-module --get-options igb

# enum_instances VMware_EthernetPort root/cimv2

You don't have vmdirectpath pass-through enabled on a VM with these NICs, right?

Someone in a similar thread said it worked for them after a firmware update of the NIC, have you tried that?: Re: Intel I350-T4 only recognized by Esx 5.5, not by 5.5 U2

Download Intel® Ethernet Connections Boot Utility, Preboot images, and EFI Drivers

If all that fails, then at this point I can only guess that maybe there is some server BIOS or NIC BIOS configuration option interfering here. Typically this could be a setting related to interrupt handling. Updating the server BIOS may help too.

Have you tried a different host or moving the NIC to a different physical PCI slot? Also you could consider upgrading to 5.5 U3 regardless of whether this will fix the issue.