What type of hardware is required for a Gigabit connection?

You should be fine with just about any hardware put out in the last decade. You're more likely to run into issues with, as you said, the NIC, the cables, or the switches. Just make sure you're using a Gigabit 10/100/1000 NIC and switch, and Cat5e or higher cables

From personal experience (I have build about 30 of these...).
For pfsense a Pentium 4 or newer is fine. If you are running more than 2 gigabit nics in there (or 2 under nearly continuous heavy load) go for at least a Core2Duo. I do recommend 2gb of RAM as that gives some room for complicated firewall, dhcp, NAT, proxy and whatever other service you want to run on the router.
I like to repurpose old business desktops for this as they often come with an onboard Intel NIC. Intel NICs are nearly always vlan capable so you don’t need a 2nd NIC for a router-on-a-stick configuration.

A hard drive is really nice. You can run pfSence from an USB stick, but booting will be slow and logging to disk will be really slow. Besides, if you need a lot of logging (or run a caching proxy) the stick will fill up quickly and may die prematurely due to excessive writing to it. (Most USB sticks aren’t really designed for that kind of abuse.)

As Tristan pointed out in a comment the upcoming 2.5 release of pfSense will require a CPU with AES-NI support. This means Broadwell Core-I (not Celeron/Pentium). Anything after Broadwell is ok. For pre-Broadwell you will need to check on a CPU by CPU basis (some Xeon, i5 and i7 models support it)