First, if they are calling you instead of you doing research on reputation and reviews and then calling them, that is a big red flag unless they are the compnay you bought it from (like Sears, say).
Home appliance warranties are like insurance - on the average over all their customers, they will pay out about half as much in claims as they receive, about 25% marketing and claims processing cost and administration, and about 25% profit and interest on their capital. Therefore, on the average, you will pay about twice you can expect to recoup in claims over the long run.
The other hand is that, if you have a large claim, you can avoid a large out-of-pocket cost by having a warranty. That means, unfortunately (from a social standpoint) that those who can least afford a large repair bill and therefore can benefit the most from a warranty (because they do not have cash-in-hand to pay for repairs) are also the ones who can least afford to buy a warranty in the first place because of limited income. People who are financially well off and have a significant emergency cash reserve can afford the repair costs, so do not need the warranty and will in the long term generally come out better off financially.
Now the caveat - for electronic devices, electronic controlled appliances like dishwashers and ranges and washers, and the like - repair cost can frequently come close to the cost of a new appliance, so sometimes it makes sense to buy the extended warranty from the manufacturer/distributor. I do this for ranges only (and it pays off handsomely because of the undersized wiring and switches they are using these days).
If you look at the consumer reports articles on this, other than trying to avoid electronic controlled appliances (most of which is needless use of electronics rather than elecrical or mechanical controls anyway), long-life items like reefers and freezers, washers and dryer, furnaces, and cheap items like blenders toasters microwaves and garbage disposals etc, you are better off without a warranty if you can afford or save up for the cost of an occasional repair. For major appliances that carry a lot of power or are short-lived like ranges and the more expensive dishwashers and clothes washers, it can pay off if they are electronic controlled.
If you buy the warranty from the manufacturer or a name-brand distributor like Sears, you are pretty certain you will get the needed service.
If you go with a third-party company like the ones probably calling you, you know they are making a lot of money somehow, and it is generally not by providing the best possible repair service. They are commonly difficult to get hold of (commonly normal business hours only, even for emergencies), have their call centers in Bangladesh or Costa Rica or similar, use the cheapest possible local repairmen (hence, serious work quality issues), commonly try to deny some or all the claim, commonly cover only parts not labor, or may cover part of the job but not secondary issues like water leaks, smoke damage, damaged food etc.
If you google the subject looking for "home repair warranty complaints" you will see there are a LOT of very unhappy customers out there. Personally, I have seen only one sort-of satisfied home or appliance warranty customer using a third-party warranty company - she got several items repaired in the first year after moving into a new home, but it was a fight every time with them on getting a repairman out and then getting the repair approved, and she was not going to give in - after 4-6 calls she was able to browbeat them into honoring the warranty - most people would not have been so persistent. Every other nieghbor I know with an apiance or home warranty (most of whom inherited the warranty attached to a new home purchase) eventually just gave up even calling for repairs, either because of the hassle or because the repairmen caused more damage than they repaired.
Air conditioners are sort of a special case - if I needed one, I suspect I would find a long-time local service company I liked whose owner also works in the field, then but into their service plan for routine sevice and repairs and labor not covered by warranty.