What is the best bait to trap the North American Gray squirrel?

I don’t know about best, but they definitely scarf down the dried corn that I leave out for them. As well, they eat bird seed.

Any leftover food from dinner they will eat. They’re dumpster divers and eat pretty much anything.

You can use peanut butter but put a cracker on top and push down so the peanut butter is glued to the little metal tray. The rodents are sly and dextrous and will often extract the peanut butter (if it is alone) without springing the doors.

Peanut butter, corn kernels, striped sunflower seeds. “Marinate” the trap in a garbage can full of bid seed if you have it. That’s how I store all my traps.

I know you are being a gentle, kind soul but in most states it is illegal to trap and relocate wild animals. I’m sure you have a good place to relocate them. Squirrels travel for at least a mile if their territory is too crowded. Be aware that you might be giving someone else a headache.

When I relocate squirrels, I relocate them to another dimension.

I “marinate” the traps in a garbage can full of bird seed not bid seed.

A method that most gardeners and bird enthusiasts would agree on is an accessible bird feeder.. Getting to the science of the question would be that they have a high-protein diet based on mostly nuts from the hard-wood deciduous trees of North America and other temperate hardwood regions of the world (other subspecies). I live on a lot with many black walnuts trees (northern Virginia) and they are more than plentiful. Fallen walnuts with husk penetrating bite inflictions are more than a common occurrence.
The success and probability of eliminating squirrels as a nuisance is pretty slim. You can maybe move them away short-term, but if there is a channel of woods (fragmented or intact) they will be back soon enough. Its probably best to enjoy them as a piece of the natural setting, leaping from trees and whatnot, or find more probable ways around.