My question to Hyatt Customer Service:
"I am currently registered to receive Free Nights under the Next Big Thing promotion. Will my stays still qualify for Free Nights if I ask for airline miles when I check-in?"
Their reply:
"According to the promotion's terms and conditions, to receive one (1) free night you must be a Hyatt Gold Passport member, register for the promotion, provide your Hyatt Gold Passport account number at reservation/check-in, be a registered guest physically occupying the room to which you are registered, and pay an Eligible Rate. Requesting points for your stay is not required."
Seems to me like they are going out of their way to keep this ambiguous.