The Reef Bay hike has some of the best views in the Caribbean. It starts at Center Road and travels down hill to Genti Bay (part of Reef Bay), which is not accessible by car so it is very deserted. Along the way, you should take the Petroglyph side trail and at the bottom there is an abandoned sugar mill.
Genti Bay is very nice and the snorkeling is fairly good as well. Of course, that means you need to carry gear along with plenty of water.
You say you do not want crowds (that is a relative term on STJ) so you might not want to take the Parks guided tour. If you do, they provide a boat ferry back to Cruz for a nominal fee (arrange details at Parks office in Cruz). If you do not take the tour, you will need to arrange travel to and from Center drop off as well as be prepared for the hike back up the trail.
Depending upon your definition of crowded beaches, most are fairly empty if you avoid Trunk, Solomon and Honeymoon after about 11 AM when cruise crowds come to visit. We just moved East to Cinnamon, Maho and Francis where the 'crowds' never exceeded 20 people. If you have a 4WD car and want empty beaches, try Lameshure and Little Lampshure on the South East side.

Cruise ship passengers go to Solomon and Honeymoon? I wasn't aware of this. The one time I went to Honeymoon (on a weekend in November) there were no more than 10 people on the beach.

You can take the Johnny Horne trail from Leinster Bay to Brown Bay. 99% likelihood there won't be another soul in sight. If you were to continue on the Johnny Horn past the cutoff down to Brown Bay, you'd come down into Coral Bay by the Moravian Church.

Any beach remote enough to get away with nude bathing??

Technically nude bathing isn't allowed anywhere, but there are probably several beaches where you could get away with it at the right time of day. Salomon (near Cruz Bay, have to hike to it) is generally considered a nude beach, but again, is not legally nude. For the real clothing optional enthusiast, I would recommend one of the French Caribbean islands, St. Martin in particular.

I went on a hike yesterday (Caneel and Margaret Hill trails), and stopped at Honeymoon Beach (right between Salomon and Caneel Bay). We were there from about 5-6:30 in the evening. Beautiful time to be there with just three other people sharing the beach with us.

Carib
Thanks for the input, we are not real bathing optional enthusiasts, we had been to Salomon beach on our honeymoon 5 years ago and tried it out, there were only 4 people there but all nude so we toss the suits. Seclusion is more important on a beautiful beach. Take Care
Linda