Mike, welcome to FlyerTalk.

The short answer is that to earn Mileage Plus miles for stays at these hotels, you have to join both hotel programs, book an eligible rate, and chose to have your points credited to United as a partner earning program. Details on the T&C pages of the respective websites. Hyatt's, for example (https://help.hyatt.com/en/hyatt-term...att-terms.html), is as follows (excerpt):

II. Earning Credit with Partner Programs

(a) Electing Partner Loyalty Points. When paying an Eligible Rate for a night at a Park Hyatt, Grand Hyatt, Hyatt Regency, Hyatt, Andaz, Hyatt Centric, The Unbound Collection by Hyatt, Hyatt Place, Hyatt House, Hyatt Ziva, or Hyatt Zilara hotel or resort, Members can – in lieu of points – select to earn miles, points, or credit (“Partner Loyalty Points”) in the loyalty program of other companies with which Hyatt contracts. “Partner Loyalty Points” is used in these Terms to refer to all partner currency in a partner program, which may be referred to in the applicable program by any name or designation, including, without limitation, miles, kilometers, and points. (The term “Partner Loyalty Points” does not include M life Rewards Tier Credits, which are discussed in Appendix D.) To earn Partner Loyalty Points in a partner program, you must be both a World of Hyatt Member and a member of the applicable partner’s loyalty program at the time of the stay. Refer to world.hyatt.com for a complete list of participating partners.

Eligible rates are rates booked on Hyatt.com or with full service travel agents. Booking.com is specifically listed as an ineligible rate.

I'm pretty sure ihg i(Intercontinental Hotel Group, Holiday Inn's parent company), is the same or at least very similar.

You will have to decide if the savings at Booking.com (if any) outweigh the benefits of earning points with UA.

Hope that helps a bit.