I suspect that you're right and that you'll need a "non-free" driver. Having a total guess, I reckon that you'll need this one: firmware-bnx2.

Ideally you should probably diagnose it properly, which I can't step you through (as it's not something I've done enough to remember OTTOMH). But if you keep in mind that TurnKey is based on Debian (v14.x = Debian 8/Jessie, upcoming v15.0 will be based on Debian 9/Stretch) then google will probably head you in the right direction.

If you want to follow my stab in the dark, then you can download the firmware-bnx2 package from one of the sites here to a USB stick. Then insert the USB into your PC. Check what device it loads as:

fdisk -l

It should come up as /dev/sdX (where 'X' is a lowercase letter, e.g. /dev/sda, /dev/sdb, etc.). If it's partitioned (which is likely, even if it only has one) you should see that and it will have a digit at the end, likely a '1' (e.g. /dev/sdb1). Once you've worked that out, mount the USB like this (I'm using '/dev/sdb1' as an example):

 mount /dev/sdb1 /mnt

Then install the firmware package like this:

dpkg -i /mnt/firmware-bnx2_0.43_all.deb

If you do the diagnosis properly (as suggested above) and end up with a different file, then substitute the name of the deb file you have. If you can't find a Debian "non-free" firmware package that supports your hardware, then things get messy and I don't have any solid advice. Although AFAIK other than particularly new hardware, most Broadcom chips should be sorted relatively easily.

Hopefully it should now "just work", although a reboot for good measure is worth a try if it doesn't.

To unmount your USB (only needed if you don't reboot):

umount /mnt

Sorry it's fairly vague, but hopefully that will at least head you in the right direction. If not, please do some googling and if you need further help, post back with whatever info you find and I'll try to assist further.