I'd get it set up. Squiers can definitely play well and sound fine. Splurge on a set up and a set of strings and play it for a few years.
What amp are you using? If you aren't satisfied with the sound, I'd save for a new amp before you upgrade from a jazz bass. Squiers aren't THAT bad, especially if they are set up properly.
You should learn how to set it up yourself though, it's really not that hard. Just google it and be careful when you are adjusting the truss rod, that's the most delicate part. If you fuck any of the other aspects up, they can easily be fixed. Trial and error.
You definitely do not need to get new frets or get them crowned, and I don't know why you'd need a new nut unless you can't get your desired action height, in which case you could file it down anyways.
Just get it set up or do it yourself, and get new strings. You'll be able to play it for years. I started out with an ibanez i bought for 100 dollars when I was 12, and didn't appreciate it as much as I should of. I've been playing Carvins for 6 years now after trying almost everything, but when I go to guitar shops im still floored with how well the cheaper basses play and sound when set up correctly.