Which USB 64GB flash drives are compatible with Mac Mini?

Any USB flash drive hardware is compatible with Macs, and stores are full of them. At this size you might not want the slowest, but the faster ones are getting quite expensive.

For use with both OS X and Windows you need to format it correctly. From the OS X internal help system:

Format a disk for Windows computers

You can also partition a disk using the Master Boot Record partition scheme, which is compatible with Windows.

I'm assuming that when I connect the USB to the Mac it gets formatted for Mac. Then I follow your directions and it gets formatted for Windows, in addition to Mac. I'd choose ExFAT since it is a 64GB USB.

No no. A brand new drive will be formatted as "something", most likely something for Windows, and that could be FAT (which a Mac can also read and write to) or NTFS (which a Mac can read but not write to).

So, you'll need to make sure it's formatted as ExFAT, instead of what it is. A medium can be in always only one format.

About the expense: Fry's is advertising a 64GB Patriot USB Flash Drive for $22.99. Is that considered expensive for a USB flash drive?

That'd be dirt-cheap, I'd say. Expect it to be very slow.

I've read answers that are somewhat related to my question, and some warn that USB's are slow. Maybe I should consider another option?

USB is not that slow (though there are faster buses). And USB flash drives come in different speeds, where especially the "write" speed can vary hugely.

The advantage of USB is that it's widely spread, and lots of media is on the market, making it very affordable. The only other affordable alternative is an external HDD, I'd say, where you get much bigger drives for less money than a small 64GB USB flash drive...

My main objective is that, in switching from Windows to Mac, I want to be able to access Windows files that I've backed up to a Seagate Free Agent. I bought a LaCie with the Mac Mini, but will still want to use the Free Agent. I need to reformat the Free Agent for the Mac (hopefully for Mac and Windows; while learning to use the Mac and Numbers and Pages I may have emergency jobs that I'll need to do with Excel and Word on the PC), so my plan is to move the files on the Free Agent to the 64GB flash drive, reformat the Free Agent so that, hopefully, the Mac will read the files on it after I move them back to the Free Agent from the USB.

Why don't you copy the content of the Free Agent meanwhile to the Mac's HDD? Then copy it back after re-formatting the Free Agent? Why would you need an extra USB drive?

Any ideas you have about how to achieve this objective will be appreciated. Maybe I should buy a third, small, external hard drive? Maybe there is a way to move the Windows files from the Seagate to the LaCie?

Exactly... you seem to have already extra space on the LaCie drive...

My understanding is that the Apple Store does not want to move files from an external hard drive to the Mac.

?????

I don't know where this could origin, but that's as wrong as it is funny. What has the App Store to do with you moving some files from one drive to another?

I should add that I don't have enough remaining free space on the PC hard drive to hold the files on the Free Agent, so that is not an option.

No, but you have a Mac with a built-in HDD, and you got the LaCie drive. Again, the Mac should have neither a problem to read from the Free Agent, nor to write to the LaCie.

From what you're telling here, you have lots of space already with what you have, I can't see the need for the USB flash drive, at least not for the purpose you mention here.

It seems that your understanding is that (a) a drive comes in a particular format for, or not for Mac, and you can't change it, and (b) that for some reason (App Store?!) Macs aren't allowed to copy files from external files. Well, both is wrong...