There have been some traditional choices in jazz guitar gear over the years. For example, jazz guitarists have favored hollowbody guitars like the Gibson ES-175 or semi-hollowbody guitars like the Gibson ES-335. However, it's safe to say that there is no one "right" kind of jazz guitar, though chances are you won't see your favorite jazz player using, say, a Gibson Flying V anytime soon. Many players, myself included, favor solidbody guitars for jazz.
The Fender Telecaster is a common choice for jazz guitarists, believe it or not (it's my jazz guitar of choice personally). If you use the neck pickup with the tone control rolled back a bit, a Tele can get a great jazz tone. If you have a humbucker in the next position, all the better! Other common solidboy choices include the Gibson Les Paul, Gibson SG, and even Strat-style guitars.
As with guitars, there are traditional choices that jazz guitarists have favored over the years. Since jazz players tend to opt for a clean, warm tone over a bright or overdriven tone, the amplifiers reflect this preference. Common amplifier choices include the Roland Jazz Chorus, Fender Deluxe Reverb, the Polytone Brute Series, and other offerings from Fender or Mesa Boogie. Depending on the player though, the rules can go out the window. Noted jazz guitarist Larry Carlton and sometime-jazz player Robben Ford have both favored Dumble amplifiers for years, and the Kemper Profiling Amp is a common favorite due to its capacity to emulate (or "profile") any amplifier ever made.
Compared to rock, blues, pop, or other styles, jazz players tend not to use many effects, though there are of course exceptions. Fusion-style players are more open to effects than traditional big band style players. Depending on the style involved, common effects choices can include chorus (just ask Mike Stern), delay, reverb, and overdrive. Volume pedals are a common pedal for jazz players as well, though they usually keep the wah pedal at home.
When it comes to other guitar gear, there is a great deal of variety amongst players. While there are guitar picks, for example, specifically named for jazz (e.g. the Dunlop Jazz III) there really is no preferred pick for jazz guitar. When it comes to strings, there is a bit more uniformity. Some players still favor flatwound strings for their mellow, warm tone and their capacity for reducing (if not eliminating) string noise, while many players prefer roundwound nickel strings, such as the D'Addario Blues-Jazz Rock sets.