Can you default a token to an irregular shape?

I am creating a puzzle for one of my D&D games, and I've been trying to figure out how to give puzzle pieces to my players so that different players will get several pieces and they have to try to fit them together on the map. I've tried handouts, and that doesn't quite work. I've also tried tokens and that is better--I can assign a token to be in the journal of a player and let that player control it, but every time the player pulls it out of the journal, it defaults to a "regular" size (e.g. a perfect square or a perfect circle). What if I want a token that is actually shaped like a jigsaw puzzle piece? A little longer on one axis than another and with irregular sides. The closest thing I've been able to do is create an image with a transparent background and make that image much larger than the actual picture. So I can show the image representing the piece, but the actual token is several orders of magnitude larger than the actual piece. That's just not satisfying. I only know how to default size tokens from the tool within the linked character sheet (1, 2, 0.5, etc., etc.). Is there some other way that you can run a macro or create an API to give a token irregular dimensions that won't change from the time you create it to when a player pulls it out of his/her journal later in the game? Thanks!

yes. Some of those ideas should/would work.The last 2 paragraphs (transparent margins around the image, or an API that automatically resizes it each time), those are both back-up options to investigate if necessary. In the first place tho, you should be able to do this easily with regular Roll20 behavior, just a matter of: Put the token on the Page (map), Resize it to the unusual porportions or aspect ratio that you want (drag the corners, or use Rightclick>Advanced>Set Diminsions). Have that properly resized token selected. Open the CHARACTER (the character sheet, you called it The Player tho), hit EDIT, then hit Set Default Token.  It will give a message that it's the token that's currently selected. Done! The one that's been coming out Square is probably because you uploaded the image as that Character (sheet's) Avatar, rather than the Set As Default Token method that's needed for it to retain characteristics like-so. However if it still doesn't work & forces it to drop in a square shape then we could look into the other ways

I think there might be a small misunderstanding here. Tokens are always a rectangular shape, whether that's a perfect square, a tall rectangle, or a wide rectangle. The token's top will always be equal to its bottom, and its left will always equal its right. You can put whatever image you want in there, and it will squash and scale it based on the final aspect ratio of the token. So, if you put an image that is 60px X 60px on the VTT and then make the resulting token 60px X 30px, you'll wind up with an image that is squashed vertically. By the same token, if you have an image with dimensions of 60px X 30px and size the token that is created when you put it on the VTT to be 60px X 60px, it'll stretch vertically.

Scott i don't think he wants a Trapazoid sized graphic... I think he wants a graphic that IS square, to come out as Rectangular every time it's re-dropped onto the tabletop.  (That's how I took it)

Also, don't bother with the Token Size attribute on the NPC character sheet. That probably shouldn't even be player facing. AFAIK, it only is used by a Compendium drop and nothing a user enters there has any effect. Follow Gold's advice above if you want to set a custom size.