The evil empire has boarded the ship. Spots are getting worse.

Help me Obi- Squash Kenobi, you're my only hope

Couldn't find anything useful when you first posted and have not seen personally seen spots like your pic - but if they are getting worse, google "bacterial spot" and see if it matches.

I looked at bacterial spot I can't find anything that looks like what I'm seeing on my squash leaves. Most of the bacterial spot photos I've seen show brown, yellow and maybe some dark black spots. Maybe they'll turn in to dark black. I hope the plant is mature enough not to go to the dark side.

Could I be a bother and ask to see the undersides please?

It does look vaguely like the first signs of downey mildew; it presents differently depending on the strain and the plant variety, but the spots are always contained within the leaf‘s veins, which doesn’t look to be the case in those photos.

Here’s the underside. The black spots look about the same up close but slightly more prominent on the front.

Wild, wild guess - fungal spores? You already treated with neem, have you reapplied?

Thanks. This is a guess too, but they look like tiny galls, probably from insect activity. Not the start of any disease. Are most of the leaves affected?

I don’t think you should worry too much just yet.

It's mostly the leaves furthest out on the vine and newer leaves closest to the base of the plant

Did you ever find an answer for this issue? I’m in zone 7a and having the exact same problem.

Sorry, but is there actually a "problem"? Boy, if faint black spots are your only "problem", you're doing great!