Hey AFGuy! I am in zone 9, in Arizona.
I would definitely get the lettuce and spinach planted ASAP, if you want to eat any of it. When it gets too hot little pests will either eat it all or it will bolt (bloom and go to seed) and be very bitter and most likely die off anyway.
The tomatoes and peppers will need plenty of root space and are both heavy feeders, meaning they will need lots of organic matter and fertilizer, so try not plant them too close together. They both like lots of sun and the peppers can tolerate the higher heat and longer hours of daylight, so you can plant them in the sunniest parts of the garden.You can start the seeds for them indoors right now and they'll be ready to go in your raised bed when they have a few pairs of true leaves (not the tiny seed leaves which look totally different than the plant's mature leaves.)
Do some research on the types of tomatoes you have to find out how much space they each need. This will depend on the variety and can differ a lot. I'd give the peppers about 1- 2 square feet per plant. At some point you're going to need to cage and/or stake the tomatoes and probably cage the peppers, as well. So keep that in mind during your space planning. Decide where you want the bigger plants first. This will save you lots of headaches later.
If you're growing head lettuce you can leave some extra space between the peppers and tomatoes and direct seed the lettuce now (while the other seeds are starting in the house). Once you get the tomatoes and peppers in there they'll provide a bit of shade for the lettuce and can extend your grow time on them a little bit.
The spinach is not really going to tolerate much heat, so I'd plant in the front where it will get lost of sun while it still cool and you can access it easily to harvest the leaves continuously without climbing into the middle of the dern box. If you're wanting to eat the lettuce as baby lettuce, plant it the same way so you can harvest the tender leaves easily and often. It will also be easier to pull them up when they begin to flower.
Squash are vines and will do very well in your weather conditions there. Maybe better than you will want them to!
You might want to consider giving them their own bed. The zucchini and yellow squash will grow well together and the vines will become inextricably interwoven. You'll be able to tell the ripe squash apart easily. You could easily do 3-4 plant in four square feet, but do them as much room as possible, they will use all the room you allow them and are generally heavy producers.
If you must plant them in the bed with tomatoes and peppers you'll want to give them something to climb up/on and will have to train them to it pretty regularly. If you neglect training them they will take over your garden and grow over your peppers and tomatoes, and might even choke them out. If I want cukes, squash, melons, what have you, I have to plant them with other things in a large bed and they do very well on a trellis that I put behind the peppers, in front of other things that will be wanting afternoon shade in the summer months. The squash will start taking off when it's getting pretty warm and the tomatoes want less sun anyway.
Good luck! Do let us all know how it turns out!