Yeah, I bet you'll have to find a third-party driver for it to work. Maybe if you can find this driver: bnx2x for it.
Also this site:
Debian HCL - HP / Proliant ML350p Gen8
Does not list the ML350 G4, but it does for G3... So I feel like it should, maybe if you go to broadcom site they may have a driver for Debian?
Do you suppose if you were to buy a new NIC card you could get drivers for that card and use it instead of the on-board NIC?
EDIT:
Maybe try copying this code and running this script?
https://github.com/demonccc/lnxDellRepo2deb/blob/master/mod_scripts/dell-ie-nic-broadcom.sh
I'm sorry you are having this issue. It is so hard to find a straight answer online anywhere I am looking
2nd EDIT:
Also try this link:
https://packages.debian.org/squeeze-backports/firmware-linux-nonfree
This package includes the Broadcom driver you are looking for I believe: Broadcom BCM5705 TSO firmware (tigon/tg3_tso5.bin)
Also from other forum:
"You might need non-free firmware for the broadcom ethernet.
These firmware blobs are available in the firmware-linux-nonfree package
Broadcom BCM5703/BCM5704 TSO firmware (tigon/tg3_tso.bin)
Broadcom BCM5701A0 firmware (tigon/tg3.bin)
Broadcom BCM5705 TSO firmware (tigon/tg3_tso5.bin)
Broadcom BCM57766 firmware (tigon/tg357766.bin)
Atheros wifi chipsets usually don't need any firmware if they're supported by the ath5k or ath9k drivers - so if it's working it's all good.
Devices supported by ath9k_htc and carl9170 drivers require kernel firmware which in debian is provided by the firmware-atheros and firmware-linux-free packages"