Is the cheapest square-shaped box this one for $13.75: https://store.usps.com/store/product/shipping-supplies/priority-mail-medium-flat-rate-box-1-P_O_FRB1 Thank you, friends :) A smaller one would be fine as long as it was similar shape. 

This smaller one? https://store.usps.com/store/product/shipping-supplies/priority-mail-small-flat-rate-box-P_SMALL_FRB  This small one isn't a similar shape as the medium.

The small FRB was designed to hold a VHS tape, if anyone remembers those.

There are two small ones about 8x5, the one linked above is 8 11/16 x 5 7/16 x 1 3/4, described as 3 stacked DVD cases. The second is 8 3/4 x 5 9/16 x 7/8 or as one or two DVD cases.

There is also another small package shaped like a shoebox: 7 3/8 x 5 1/4 x 14 7/8

https://store.usps.com/store/product/shipping-supplies/priority-mail-shoe-box-oshoebox-P_0_SHOEBOX

Another small one almost cube shape: 7 1/4 x 7 1/4 x 6 1/2

https://store.usps.com/store/product/shipping-supplies/priority-mail-box-4-P_O_BOX4

I have never had a problem ordering in these since they're free. If I have to get more than I need I take the rest back to the PO with no problem; I suppose you could put them in the paper recycling too.

Are they flat rate boxes, or cheaper to mail than flat rate?

My apologies for not replying until now. I appreciate all of your input :) I'm going with the medium. 

Diane,  they are flat rate (if it ships, it fits) boxes and envelopes. I've often used the flat rate padded envelope for Christmas and other gifts. Just as an fyi, you can order free flat rate envelopes and boxes through the usps.com Hope this helps:)

The boxes linked above are not flat rate. If you use PirateShip.com to ship them, they will be cheaper than a Medium Flat Rate box.