The sawdust-looking stuff (technically called frass) is definitely vine borer larvae working through your stems, and once they're in there it's basically game over for that plant. The good news is you can still save your current crop and prevent this next year.
Here's what actually works without chemicals: if you catch wilting early enough, you can sometimes dig into the stem with a small knife or wire and physically remove the larva before it does too much damage. I know it sounds gross, but it works maybe 60% of the time if you're quick. Once you extract it, mound soil over the injured section of the stem - squash plants can actually root along the stem, so burying the wound gives it a shot at surviving. The bigger play though is what you do going forward. Plant your squash earlier (late May/early June where most people are) so they mature before peak borer egg-laying season in mid-July, or go the opposite direction and plant a second crop in late July that won't be vulnerable until fall when borers are less active. Some folks also wrap the base of stems with aluminum foil or old pantyhose as a physical barrier - sounds janky but it actually keeps egg-laying insects from reaching the soil near the stem.
The hack nobody mentions: plant peppermint around your squash patch. Borers don't like it, and you get free mint. Also, this might sound counterintuitive, but let your squash plants sprawl messily rather than training them neat - dense foliage makes it harder for borers to find entry points. And yeah, clear out plant debris in fall because the larvae overwinter in dead stems, so don't leave last year's squash plants rotting in the garden.