Some books recommend a gentle bounce when bench pressing or squatting. This feels wrong to me. It feels safer and more controlled to come to a complete stop, then start up again. What should I do?
You should follow the advice in the book that describes the program you're doing. Unless you find a reputable source that describes specifically why bouncing is dangerous, don't worry about bouncing out of the bottom of squats or bench presses.
It's "cheating" a little to bounce in the bench, so for optimal strength gains you would avoid that. For the squat it's totally normal.
Bouncing in the bottom of a squat, done properly, is an effective way to maximize muscular output, as described in Starting Strength on page 138:
...a key feature of efficient squatting is the use of the controlled "bounce", which takes advantage of the stretch reflex that occurs at the transition between an eccentric and a concentric contraction. Any muscular contraction is more powerful if it is immediately preceded by a stretch, as always occurs when you jump.
If you don't want to bounce because you don't know how, squatting with no bounce at the bottom is significantly harder, but a reasonable option. You won't be able to lift as much, so you won't get as strong, but it's still squatting.
Bouncing put of the bottom of the bench press isn't dangerous (you're not going to be suddenly lifting heavy enough to drop a chest-crushing weight onto yourself), but it's cheating.
Bouncing out of the bottom of the squat is making use of the stretch that is applied to your hamstrings at the bottom position and is recommended, at least by Starting Strength. At heavy weights, there's no way this will actually be an observable bounce, but you should think of it as such.
For bench press, you get more benefit with touch and go (not a bounce off the chest). If you ever plan to do competitions, you will also have to train paused reps (1s pause on the chest).
The squat has a stretch reflex built in, and a controlled bounce helps complete the rep. The important word here is controlled. You don't want to loose tightness because of the bounce.
In general, if you bounce the bar at all, do not lose tension. Once you do, you have minimized the strength you can apply, or worse made yourself prone to injury.