The "Borg nanoprobe" seems to be the Borg's most powerful weapon. The Federation and others obviously have similar technology but have they ever copied, developed or controlled Borg nanoprobe tech? They've had plenty of opportunities to acquire samples in order to back-engineer the technology.
As Xantec noted and Richard provided a link in the comments above, Voyager used modified Borg several times:
In 2374, The Doctor was able to modify nanoprobes as an offensive weapon against Species 8472. He reprogrammed them to mimic the alien cells' electrochemical signatures, so that they could evade detection by Species 8472's immune system while continuing to assimilate. These nanoprobes could be delivered inside a photon torpedo or along a phaser beam. (VOY: "Scorpion", "Scorpion, Part II", "Prey")
Nanoprobes could also be modified to attack specific cells which contained a virus or some other disease and so help the patient recover. In 2375, Seven of Nine encoded some of her nanoprobes to assimilate the synthehol molecules in Kadi Ambassador Tomin's bloodstream, allowing him to recover from his intoxication in time to return to his people. (VOY: "Someone to Watch Over Me")
Since bio-neural circuitry is organic in nature, nanoprobes could also be used to influence or disable this type of circuitry. (VOY: "Warhead")
They have... on several occasions in Voyager.
They used Seven-of-Nine's nanoprobes to revive Neelix when he was killed by a protomatter cloud discharge in VOY: Mortal Coil
They again used her nanoprobes to 'arrest' the alcohol particles and make an alien ambassador... well, sober in VOY: Someone To Watch Over Me
Another occasions saw the use of Seven's nanoprobes to bypass/repair Iko's edema in the brain (Which also seemed to have activated his conscience) in VOY: Repentance
Apart from that, her nanoprobes were also sampled and replicated on several occasions to be used in effective manner against species 8472 in VOY: Scorpion
Also, the Federation DID develop their own nanites in medicine for surgeries, etc in TNG... however, this is not frequently seen (otherwise, the Doctor in Voyager would not have resorted that often to using Borg nanoprobes).
Then again, none of the show's doctors (either the EMH or Bashir) were seen using stem-cells to regenerate lost limbs in people (Nog's leg for example) or rejuvenate the body to reverse ageing which is already largely possible in real life. Trek seemed to have no problem with prosthetic limbs, etc. but no use of actual stem-cells.
I'd imagine that when it comes to program-ability in regards to medical aspects, Federation nanites and Borg nanoprobes are pretty similar if not downright the same. Actually it would be easier and far less dangerous for the Federation to use its own medical nanites than Borg tech which could technically assimilate a person if something goes wrong.
Granted, Borg technology seems to be more adaptive, but the underlying basis of their technology is nanotechnology... and the Federation developed its own version before meeting the Borg... so there's no reason they couldn't use their own nanites and possibly use some of their experiences with Borg nanoprobes to enhance function of their own nanites (not just in medical aspects, but others as well).
CRUSHER: Nanites. Tiny machines built from the atom up. Designed to have exposure only to the inside of nucleii during cellular surgeries. Until then, they are kept tightly confined in a non-functioning state.
DATA: These are not ordinary nanites.
CRUSHER: No, they have evolved.
STUBBS: Evolved? How does a machine evolve?
WESLEY: It's true. I am responsible for this. I allowed two of the nanites to interact for a school project. I wanted to increase their capabilities. And they escaped.
The Federation seems to be using their own developed nanites in cellular surgery, however, prior to Wesley allowing them to interact, they have been using them in a fairly limited fashion. Also, Beverly Crusher DID suggest introducing a destructive breed of nanites into the Collective at one point.
However, I find it very unusual (not to mention implausible) that the Federation wouldn't have the technical ability to program the nanites to interact with each other inside human body or outside of it and perform specific tasks without 'spontaneous evolution' occurring.
And before someone says that this might be akin to keeping nanites in a state that makes them not self aware... one could argue the same thing about Holodecks too, because they are also capable of creating fully self-aware AI for example, but that still doesn't prevent the crew from using the holodecks for training, relaxation, recreation, etc.