I'm hoping an expert comes on here, but as I understand it, you can receive "stuff" that allows you to participate. We had an identical pitching outing (kept the hat) and I marked the equipment/apparel box while indicating zero cost. Had other instances where some program fee was paid, and received equipment/apparel in those instances also. I think you can play for free, even up to having your travel/lodging expenses reimbursed is still retain your amateur status. Read in places that you could even receive "winnings" as long as it did not exceed your actual expenses. The instance where all you keep is the hat seems harmless enough, but what if they also handed out one or two wooden bats that you got to take back with you. At what point does it rise to the level of checking the box? I choose to take the conservative route and went ahead and checked the box anyhow along with entry fees. PG has a logical point about individuals not paying, but logic may not apply here.
In looking at the actual NCAA regulations, there seems to be an almost infinite amount of exceptions to their rules. One of the more interesting ones was for amateur tennis players. Apparently they can keep up to $10k in winnings and retain amateur status. After the $10k is reached, they can still accept prize money up to their travel expenses. Sounds like you need to make hay during the summer and then pick some nice locales during the winter and get the airline ticket and suite paid for.