For a commercial airliner, how strong and reliable are the vertical and horizontal stabilizers and rudders? Can they just break off?
They are very strong and reliable. Occurences where they are completely broken off are extremely few and far apart, and in the cases that have happened, usually some kind of abuse took place, such as too harsh control inputs from pilots. In the case of American Airlines flight 587, the control system was also a contributing factor.
Lesser damages, such as cracks and deformations are also quite rare, an example of the more serious kind being the VARA ATR 72 flight during which excessive control inputs damaged the tail assembly.
As anything in airplanes, they are not designed and built to endure everything, as such a design approch would lead to very heavy and/or expensive planes.
Safety is absolute paramount in aviation. There is no industry in existence that has a higher safety standard than aviation. Aviation operates so called idiot proof systems. Routines that completely eliminate the possibility of failure. Pilots are never ever done learning to fly safe. In highly realistic simulator flights, pilots are taken through any problem they could expect to encounter without prior warning. By being forced to go through their own simulated death again and again, they obtain a mental condition that allows them to remain mentally operational under the most insane circumstances. Aviation rules and regulations is the only set of traffic rules that in so many words obliges to ignore all its other rules completely if such is necessary to ensure safety. The safety standard of aviation completely dwarfs the safety standards of for instance health care or nuclear industry and for good reason. Mishaps in aviation tend to go real bad, real fast.
This safety standard also reflects in the design of any and all aircraft parts, including control surfaces. Though there are many cases of aviation disasters involving failure of control surfaces, there are also cases in which partial or even complete loss of horizontal stabilizer and rudder control in mid flight proved survivable.
So do control surfaces ever 'just' break off? There is to my knowledge no relevantly recent record of any case of an airliners fully functional and properly maintained and handled control surfaces spontaneously breaking of in mid flight. That covers the 'just'.
For rudders or stabilizers to come off, something else must go wrong first. Poor or incomplete maintenance, incorrect training or so far undetected design flaws may cause that. Bad weather may not. Some people fly in and out of full blown hurricanes for a living, without considering themselves to be stuntmen in any way. Turbulence strong enough to make you break your neck banging into the ceiling of an aircraft unless you had your seat belt fastened, will not rip off the stabilizers.
So can they ever break off? Theoretically, maybe, realistically, no. The odds of that ever happening at all, limit to zero. The odds of it ever happening to you are so small, that the mere influence of you thinking about it becomes a significantly contributing factor.
On August 12, 1985, a Boeing 747SR operating as Japan Airlines Flight 123 lost its vertical stabilizer and crashed after the aircrew had struggled for 32 minutes to control the stricken airliner. It is the deadliest single-aircraft accident in aviation history; 498 of 502 aboard were killed.