Why would the neck need to be replaced? It's still there and still holds it's shape fine. Does it get old or worn?
The frets are noticeably worn down with little valleys under each string, but they are still elevated enough that they are causing no problems. At what point to they need to be replaced?
The nut appears to be plastic. It looks like there is a 1/16" layer of something clear on the back of the nut. Not sure if it is a glue holding it in place, or if it's a layer of lacquer used to elevate the nut. I'm hesitant to rip it out until I do a little more research.
Yes the squire was cheap. I think I paid $99 for it back in the day, and it had a "list price" of $299. I have played other squires over the years, and i've always liked mine and felt it was better quality than the other models I have played (mine does not seem like a high end model at all, just 1 notch better than the cheapest squires i've played). Still, she's my girl and we have bonded over the past 16 years.
It's probably lacquer holding it in place. You want to use a razor to cut the lacquer before you tap the nut out, otherwise it might chip around the wood.
The neck will last pretty much forever provided it doesn't warp or have the truss rod break. Typically with bolt on neck guitars it is cheaper to buy a new neck than have it refretted though. You can do a refret yourself . . . but it's a very time consuming and finnicky process (more so with maple as it's finished, so you have to repair the finish after the refret too). It's possible to screw up the neck pretty badly doing a refret, (you can introduce backbow by going with too wide a tang on the frets) and after your done you have to properly level and polish the frets.
Frets wear slowly, so it often creeps up on you in very tiny increments. If your frets are very worn you'll notice little things . . . like you have to press much harder to prevent a note from buzzing, it will be very difficult to do fast pull-offs cleanly, barre chords require more pressure, tapping passages become nearly impossible to play, the divots in the frets will actually change where the string sits so your intonation will go off, stuff like that. The biggest problem for me is the extra pressure you have to use to fret . . . it ends up leading to poor technique (you get a death grip on the strings) and all the tension can lead to wrist problems when you're regularly playing two hour sets.
Those tusq nuts usually work great, and should sound better than the cheap plastic one you had before. Just fit it and check the height a few times, sanding down as necessary. DON'T use epoxy to finally stick it in place, just a tiny dab of wood glue mixed 50-50 with water will do.