"Instant Martians. Just add water." - Marvin.
There are several technological secrets that the Greys have that several parties on earth would be extremely interested in.
The Void Engineers would love to be able to master this in order to help further the cause of science by taking the Technomancy to the stars and beyond. The Military wouldn't mind it for pretty similar reason. The powersources of each races ships are completely different.
The Grays favour magnetic pulse engines that allow them to travel at great speed, making turns with out any need for slowing. The Zeta Reticulans prefer to use anti-gravity engines when in the atmosphere and use lightspeed jump engines when travelling through space. Only a few Alphans arrive on planet in vessels similar to the Zeta Reticulans, but once they settle in they are quick to set up Hyperspace Teleport/Jump stations, through which they receive their supplies, reserves, and extra equipment, as well as sending reports, personally. Of these the Military are particularly interested in so that they can teleport bombs at their enemies. Some Alphans also carry small wrist devices that "look" like watches that act as recall devices. If they are captured, they merely twist the dial and they are transported back to the nearest way station. (These are not similar to Void Engineer technology as they bend space rather than open doorways to other space.)
As stated the Greys never carry anything that can harm another creature. The Grays themselves use their mental abilities, but their ships have been shown to contain some powerful emissions that may or may not be a result of boosting their own abilities. The two that rays that are used are both blue in colour but vastly different in results. The Grays tractor beam is capable of halting a fighter jet in flight and dismantling it while it is still in the air, and their weapon ray blasts almost anything within a half a kilometre range to ashes but is only used if they feel they have no other alternative. Some ships have also been known to be able to project implant chips from a great distance away. The Zeta Reticulans are too careful to introduce such alien technology, and the Alphans are too pacifistic. However, it does not take too much simple Earth ingenuity to see the damage effect of some of the devices that the aliens carry.
The Grays keep their apparatus on their ships at all times. These tend to be a range of probes (how anime you want to make it I leave to you) and devices for studying the body, collecting gene samples, impregnating, and anything else you think they might be looking for. These arms could be used for self defence purposes if it comes down to that. The Grays use a sheet of almost living tissue/webbing that they drape over the bodies of their specimens, that "seals" them to the slab. It opens at the touch of the Grays, but never loses any strength regardless of how thinly it is separated.
The Zeta Reticulans are more than protective of bringing anything to Earth that might be discovered. The reason behind this is that anything they bring of an obviously alien nature will reveal them or, even more disastrously, be studied and used by the populace, making them responsible for a major change in Earths development. What they do bring is kept in their dwellings. These include their food machines (nasty stuff, any Earther who eats it will declare it bland and will succumb to muscle spasms a full hour after consumption, for as many minutes as grams they ate), monitors (designed to let them know of disturbances of a genetic or technological nature that might be the result of other Greys or natural human development) and a contact device that calls the mother ship in case of an emergency.
Alphans like to carry little things about their person of a higher tech level because it pleases their arrogance to do so. These tend to be little amusements or games (holographic logic puzzles, etc). At their bases they keep their subliminal technology where they monitor and study the world in general as well as maintain laboratories where they "downgrade" their technology for human consumption. Microwave ovens are a result of the Alphans. They also maintain warehouses where they store mass produced items that haven't been announced to the public yet.
Despite the huge technological leaps that the Greys have over us it is clear that they still don't have some things we've developed. While the chips in the back of necks might be incredibly small, no Greys yet have the ability to create Nanotechnology (they don't see the need for robots to fix them internally when they either live or die). This may be one of the few areas where the Technomages are capable of defeating them. (Or is it that they have had nanotech but have decided against it? or they never created the concept of nanotech, aliens that they are. Or possibly they fear the possibility of replicating the ways of the death machines.)