To replace the nut on my squire strat with a locking nut?
It'd be pretty sweet if it was.
Yes, but it wouldn't be worth it, unless you're installing a Floyd Rose-type tremolo. Once the nut's locked the strings go sharp and you can't retune without taking the nut off. Also, it'd be too much work and would be easier to get locking tuners.
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Well, you would need a bridge with fine tuners.

If you want better tuning stability, string your strings properly, get a better nut or lubricate your nut, look into better saddles, or get locking tuners.
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