I think you were correct. Their, his, my, all refer to a thing, owned by a person. That may be a physical thing (dog, hat), or a concept (skill, soul), but it's a thing, and therefore a noun. You cannot own a verb.
Here's my hobbyhorse: When a reporter on the telly is talking about something rare and describes it as 'one of the only ..... left in the country'. 'The Only' is a singular item in that context. You cannot have one of the only. You can have 'one of the few', or even 'one of only a few' (where only is.. well I dunno, an adjective?), but not 'one of the only'.