If the HDD is totally free (external caddy) and can be attached (USB/firewire) to either computer and has adequate space the answer is yes. Backing up two PCs is no harder than one; it just takes more time.
I use an external drive which I only attach when backing up. I see no reason to leave it connected continually unless you are using it continually.
My backup media of choice is Acronis. But the licence only covers installation on one PC so legally you should buy two - up to your conscience. I think most commercial software is for one installation. I am unable to suggest a free alternative as I have never used one; perhaps other will be able to help here.