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In which area of Puerto Rico?  Showers are not uncommon.

Playa Montones in Isabela has outdoor showers, but to use them you have to pay $10 to park in the resort lot and use the facilities.

Crashboat in Aguadilla has outdoor showers. They are free.

Playita Rosada in La Parguera have showers and it's free.
Caña Gorda in Guanica is $3/car have showers and cantina.
Boqueron in Cabo Rojo does too and is free.

In the San Juan area, Escambron in Puerta de Tierra has public showers, and I bet the Balneario Publico in Carolina (not in Isla Verde, but just east) also does.

"in PR" = all areas of PR

The has a list of their beaches with individual facilities on their website. Look for "ducha" (shower) in the facilities list.

One of the beaches I have visited is Cerro Gordo Beach in Vega Alta. You pay a few bucks to have secured parking. It's a beautiful beach with a huge area to choose from, including many tall Palm trees that provide shade if needed. Depending on the season, they will also have food trucks set up. Last time we went there were 4 to choose from, including a soft serve ice cream truck. You can also bring a grill if you are so inclined. I've never seen a beach where there wasn't a grill though. There are few places in the surrounding area to explore and they even have a MTB trail.

Luquillo.

Seven Seas, Punta Santiago.

Rincón Balñario (public beach) also has outdoor showers.

I was going to say the same about Rincon's beach.

Check out this report on PR beaches.

Here is some information on the la parguera area.

Sun Bay, Vieques and Playa Flamenco, Culebra have showers at the campgrounds which anyone can use as there is not any way to actually discern campers from the general public. No one cares actually as they are both public beaches with public campgrounds.

Guanica public beach on 333, next to Copamarina.