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Wikipedia article tells that supercruise is a condition when an aircraft can achieve velocities above Mach one without using the highly inefficient afterburners. But still, there are not a lot of engines which can achieve this.
What is the specific requirement for an engine to achieve supercruise? Why can't most of the engines reach supercruise without reheats enabled?
It's less about the engine and more about the combination of engine and plane. Theoretically, all that is needed is for the engine(s)' thrust without afterburners to be able to overcome the plane's inherent drag at a speed greater than Mach 1. Simple, right?
Well, to do that, you're up against some pretty heavy physics. First and simplest, given a drag coefficient for the aircraft in a "clean" configuration, the drag force increases on the square of velocity. So, to go twice as fast, you need four times the thrust with all other things being equal. Jet engines simply don't scale that way; you need a bigger engine to provide more max thrust, and then all things aren't equal anymore.
Second, planes aren't built like cars, where in many cases there's enough room under the hood to drop in a bigger engine block and a turbocharger to double power for negligible additional weight. Fighter designs have to essentially wrap an aerodynamic skin as tightly as possible around the pilot, engine, avionics and fuel storage. For the longest time, engine efficiency was pretty static, therefore more thrust meant more engine. A bigger engine means a bigger plane, which means more drag, which offsets the thrust gains and thus the speed increase of the larger engine.
Thirdly, there are some complex aerodynamic changes involved as a plane transitions the sound barrier. Airflow into the engines increases at higher speeds, which reduces the compression ratio of the engine; the intake turbines become less and less efficient as speed increases, and thus you get less and less benefit from the "bypass" airflow of the turbofan. Afterburners are more or less a simple way to continue to get more thrust after the intake turbines are no longer significantly compressing air, by taking that air, dumping a bunch of extra fuel in it and riding the prolonged explosion. This, however, uses ten to fifty times the aircraft's cruising fuel flow rate.
The F-22 achieves supercruise in part by extending the efficient operating range of the engines with "variable bypass". At lower airspeeds, engine bypass is increased to allow the turbofans to increase thrust with minimum fuel. At higher airspeeds when the turbofan effect decreases, compressor bypass is reduced to put more air into the combustion chamber (along with more fuel), which increases thrust beyond the normal compression limit more efficiently than simply burning the bypass airflow with an afterburner.
Finally, there is the simple requirement of supercruise in an aircraft's mission profile. Not many things in air combat occur at supersonic speed; you can't dogfight effectively at Mach 1 because your sustainable turn rate is measured in miles, you often can't drop gravity ordinance (bombs) because they won't fall away from the aircraft in a stable or predictable way, and in the case of the Raptor you have to open payload doors to deploy your weapons and that will increase your drag right there.
Supercruise has a singular purpose; to increase the speed of the aircraft's transit to and from the target zone without sacrificing range by dumping fuel into the afterburners at ten to fifty times "military power". For an air superiority fighter (combination interceptor/dogfighter), that's very important, but the F-22 was the first U.S. fighter developed from the ground up to take advantage of the latest technological progress in aircraft design such that it could be achieved. The last time U.S. fighter designers got an opportunity to design a "cost-is-no-object" superfighter was almost 40 years ago with the F-15, and the two major fighter designs to enter U.S. service in the interim, the F-16 and F/A-18, were the result of a design competition specifically intended to create a lower-cost multirole fighter to serve as the bulk of the air fleet.
Other supercruise-capable fighters designed by NATO allies reached production first; the English Electric Lightning interceptor is considered the first supercruise-capable production fighter (predating the coining of that term), and the Dassault Rafale is reported to be supercruise-capable in certain mission-ready configurations up to Mach 1.3, and beat the Raptor to market by about 2 years, though the Raptor has a substantally higher top speed under full military power (100% non-afterburning throttle).
Almost any jet engine can be used for super cruise. Certain characteristics help, however:
The aircraft needs to have sufficient thrust and low drag, so area-ruling and a low flight mass help.
Supercruise was invented as a marketing term for the Lockheed F-22. Note that a number of earlier designs could also cruise supersonically without afterburner. The stealth design of the F-22 requires the internal storage of weapons, which makes achieving super cruise easier. External stores full of weapons and external tanks would make super cruise practically impossible.
