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I have a couple of Squiers (jagmaster/duo sonic) and I was wondering if a bone nut would improve the tone significantly.

not likely, but i'm sure someone else will have another opinion

Not sure about 'significantly' but there is an improvement, i fitted one on my Jag-stang. The sound appeared to have more note clarity.

i had one put on my jagstang back in 95 because the low E section snapped off on the stock nut. my guitar teacher/old shop dude said they were way better at the time. i was just happy to have it fixed.

as far as open string sounds, i guess there is a difference but i think its not that great, and actually the way the nut is cut has alot to do with the way the string will vibrate.
and bone doesn't need to be lubed up for trem use.
besides that the "tone" you may be after will not be found in a bone nut.

I'd get one of them graphtech nuts if it were me.

But I wouldn't spring for a new nut until you actually need one

I have a real bone nut I made from a blank on The Colonel, and own a couple CIJ fender RIs and one AV... the cyclovac plastic nuts Fender now use (and label "vintage bone nuts") are indistinguishable from the real thing when all is said and done... I dunno if they're easier to work (because I've only done work to the real bone one), but you will not notice a difference in feel or sustain or sound.

fwiw I also have a brass nut that replaced the plastic one on my epi g-400 and didn't notice much a difference, and a graphite nut on my Bananacaster that also didn't yield much of an improvement to tone/sustain. So I now feel that talk is mostly bollocks.

If the stock nut is crappy, that's a different story. A new nut improved my SX Jazz bass's action and playability twofold.

I totally agree with paul, in my experience, the difference in tone down to material is fairly indistinguishable to my ears.
A good nut is a good nut and a bad one causes a lot of problems.

I was having issues with the Duo Sonic. when I would push down on a "D" chord the notes were going sharp.

the guitar tech adjusted the neck and it improved. I was wondering if the nut might be part of the problem.

A badly filed nut can bind the strings and cause tuning problems while a nut that is too low will cause buzzing problems and a nut that is to high will result in a high action. I don't think I've ever played a guitar that had a real bad nut issue though.

Loose holes are a bigger deal. But now that I do most of the shaping with a Dremel, I can make one in about 10 minutes. I do think in some cases, the tone was more clear. Like when you rap a tuning fork on a hard surface. But like everyone has said, it's only a small fraction of the tone. More about string movement and no buzzing.