I am a fan of the healer companion. Even at 2700 item level, I still pull my little Snowy Fawn out, whether it be to get me "just a little more" health in a difficult area, or to increase lower level characters' survivability as we work through Stronghold Heroics. And he's just so darn cute.
I have done some testing on a few of the healer companions and found that the main stats that affect their ability to heal are: power and regeneration. I suggest stacking your companion with power. Regeneration gear is not very common in NW and there are currently no runestones(used by companions) or enchantments which have regeneration on them, so you can not really "stack" a large amount on your companion. However, there are runestones and enchantments, as well as companion specific equipment, with power so you can stack power MUCH higher than you can stack regeneration. The higher the healer's power, the better it heals.
To this end, healer companions that have 3 offensive runestone slots make better healers because you can put a power runestone into each slot. I find that putting defensive runestones on a healer companion does little to increase its survivability, so you may as well make it "really good at its job" while it is alive.
As far as equipment, I won't say much here because that is for a different post, but I will mention that my preliminary testing on companion equipment showed NO increase in healing (from your companions) when using the Loyal Commander piece over the Loyal Master piece. (tested on Fawn of Shiallia, Snowy Fawn, Apprentice Healer, and Lillend)
I also did not detect any increase in healing from the recovery on the Loyal Commander piece (however, i only tested the recovery with the Spring of Life spell the fawn uses which goes down every 30 secs). I have also read from other player testing that it does not work for companions.
A caution when using healer companions: they tend to have a wide roaming area. This is because they try and stay clear of combat which means they can draw in other mobs in the area that you may not have intended to engage. Also, mobs love to chase after your little healer and it will try and run away, again, leading to the risk of unwanted attention.