There are already some notes on "heavy weather running" and "heaving to" in 'Ask the Junkmaster'. They refer mostly to two masted rigs, but one masted rigs aren't so different to handle that those notes are invalid. The main difference is that heaving to is simply a matter of hauling the deep-reefed sail amidships so that both it and the boat are in a "stalled" condition, so that the boat drifts sideways and forwards as slowly as possible, with the flat sail amidships providing resistance to rolling. Where the helm is will depend on the boat type, but it will normally be lashed to leeward. That's the only area I can think of where handling a junk rigged boat differs slightly from handling a boat with another rig. So I'll offer some more general notes, applicable to all rigs, I hope.
1. Make lots of passages, get in lots of sea-miles. Then the spectre of heavy weather at sea will tend to retreat into the background. It's always going to be unpleasant, but it's not often life-threatening, and it needn't happen all that often. It's the sailor as much as the boat that needs to be made ready for heavy weather. "Book-learnin' " will only take you so far, you have get out there and learn about yourself and your boat.
2. A cold, wet, tired, sick, hungry, thirsty sailor can't make good decisions, can't function well. Junk rig allows for setting up a sheltered watch-keeping position, with self-steering controls and rig controls easily workable from one position - make it so. Find out what digestible, comforting food and drink works for you when the going gets tough. Have a vacuum flask of hot drink made ahead of a short-lasted rough time.
3. Prepare the decks. Don't let the clutter build up. Fasten down everything that must remain on deck, because you won't want to go there when it's rough. Make sure your vents let in air, but not water.
4. Prepare down below. Put in lots of hand-holds. Take out the sharp corners that would injure you if you fell on them. Anticipate what will happen when gravity fails to work normally, and momentum takes over. Fasten down horizontal locker tops, wedge books in place, make the galley secure. What you really don't need is spoiled food, soggy paper, cooking oil and broken glass on the cabin sole.
5. The best places to be are either tucked up in an all-weather harbour, or way out to sea. Being close to land in shallow water, or in an anchorage where a wind shift can catch you out, is asking for trouble.
6. Control down-wind speed. The junk rig helps here, with it's ability to be reefed right down, panel by panel, even when running. When the sail is reefed right down, use some form of drag device to keep the speed in the boat's mid-range - not so slow that she won't steer, not so fast that she trips over her keel and broaches.
7. Stop when necessary or desirable. Heave to, to cook, rest or sort things out on the boat or in your mind.
8. What is heavy weather for a small boat is a good sailing breeze for a large one. Learn what your boat can do, and what she's uncomfortable with.
There, not so much in there about junk rig, was there? That's because it's the least of your worries. You have a rig that makes handling heavy weather as easy as it's ever going to be. Two years ago, I sailed in Tystie from Vancouver down to San Francisco in October, at the same time as a 65' bermudan cutter of very high quality, a fellow member of the OCC. It was rough. They reported having had a terrible trip, the gooseneck broke, the tracks ripped off the deck. In Tystie, it was "just another day at the office". We jogged along, adjusting the sails easily to the rain squalls as they came through. The seas were very big, but we stayed warm, dry and comfortable inside, using the pramhood to full advantage. If Tystie had had bermudan rig, and if I had had to do any deck work, I too would have been reporting a terrible trip.