I have not contracted lycanthropy or participated in any Companions quests.

However I received Hircine's cursed ring which supposedly causes you to turn into beast form at inconvenient times.

Does this have no effect if you are not already a werewolf? I've had the ring for quite a long time and never turned into beast form.

Oops misread, Yeah its not happened to me as I havent used it as Im already a lycan.

I had the cursed ring equipped (non-werewolf character) through several quests while I ignored the rest of that one and I never once changed. This isn't to say it's not possible, but certainly the chance must be very low if it is.

Can you equip the ring on a companion? Will THEY turn?

From what I understand, the ring only gives you additional transformations if you are already a werewolf, at least that's what the tooltip says.
Which means you'd be able to turn more than once a day. It doesn't give you or your companions lycanthropy by itself.

Once you receive the ring, you can't remove it. It's cursed, so that is the only ring you will be able to wear until you complete the quest.

After dying a dozen times, I decided to hang on to the ring for a while and finish the quest at a later time when I'm strong enough to survive.