What has gone wrong with my squash, cucumbers, and peas?

Winter squash are called that because they're left to mature and stored for use in the winter. They still grow in the summer. Curcurbits need a LOT of warmth and sunshine.

The squash leaves don't look mildewy to me. But the colour of the squash leaves isn't as green as I would expect, even the smaller newer leaves. Nutrient deficiency? Touch of cold weather? A quick google yields http://agropedia.iitk.ac.in/content/deficiency-symptoms-nutrients-and-their-remedies-cucumber which would point towards a possible magnesium deficiency -- you're seeing chlorosis and drop of older leaves. The peas, too -- they should be a little brighter green than they are, and they definitely don't have powdery mildew (you would see it covering the leaves). It looks like they're finishing early but due again to a possible nutrient deficiency?

Can you get a shot of the underside of a squash leaf?

The peas might just be done but the others definitely look like some kind of mildew. Mild and moist is perfect weather for it unfortunately. I have had success in the past taking a couple of leaves into my local garden store and asking what they thought it was. You need one of those good shops run by gardeners who know about local conditions and things. I'm also happy to throw fungicide or insecticide onto my plants if it keeps them happy so would generally buy whatever based on their recommendation, but even getting a clear identification can help you look up possible alternative treatments if you don't want to go that way.

Did New Orleans get hit by that odd storm system that spawned tornadoes all over the South right around Christmas (before it came up and dumped a bunch of snow on me in PA)? The squash and cukes are warm-weather crops and would've suffered if it ever got near freezing. My theory doesn't explain the peas, though.

I agree... Peas look done. Is that summer squash, or winter squash? Even if it's not particularly cold in Louisiana, it may be responding to or compromised by the shorter days. I would contact your local extension office or organic farmers association.

For the most specific regional information, you might contact your http://www.lsuagcenter.com/en/our_offices/parishes/ (there are http://en.wikipedia.org/wiki/Cooperative_extension_service all over the country, run by big state land grant universities). The LSU cooperative extension appears to have a "Plant Diagnostic Center" or "Plant Doctor" in each parish who is available to take calls, look at samples, and identify problems for you.

These guys: http://www.lsuagcenter.com/en/administration/about_us/extension/

+1 for powdery mildew, renowned for destroying cucurbits. Very sorry to say as much, but you may have to pull everything and trash it outright -- no composting, as compost pile temperatures will not kill the spores (which travel on the wind). http://gardening.about.com/od/gardenproblems/qt/PowderyMildew.htm, http://www.gardenguides.com/779-using-milk-control-powdery-mildew-garden-pest-tip.html, and http://en.wikipedia.org/wiki/Neem_oil are often used as preventative measures for powdery mildew and other fungal plant diseases, so you may want to give those a shot before giving up hope. I've tried sulfur and copper fungicides with middling success, albeit at earlier stages in the disease, and have heard good things about http://www.groworganic.com/weed-pest-control/organic-fungicide.html but never used it. To help prevent similar problems in the future, only water the base of the plant rather than using a sprinkler or watering from overhead (best practice in gardening anyway), space your plants as wide as possible, and grow disease-resistant/tolerant varieties if possible. You may still be able to wrest a squash or two from your garden, if the fruit is already fully formed and currently ripening on the vine. Any harvest you get from these plants -- which are popping out fruit because they feel they're in danger of shuffling off this mortal coil, last chance to reproduce! -- is 100% safe to eat, although it may be rather bland since the plant is highly stressed and unable to photosynthesize tasty sugars as efficiently as it should.