As per the supposed specs, it should run. I get a "no suitable graphic card found" error when I try. I also have DirectX 11 currently.

I've been having the same problem myself and have been researching solutions quite thoroughly for the past few days. From what I've read about the game not working on laptops, you may get it running by going into the "control panel" for your graphics card, then find the setting "Preferred Graphics Processor", and change it from "Auto Select" to "High Performance"

I hope that helps and at least one of us might get to play this game. :)

Unfortunately, that seems to be for NVIDIA graphics card; I am working with the onboard Intel HD 3000 Graphics Card (pretty low-end machine, Dell Optiplex 755, but works for my taste in PC games).
I believe the minimum requirements were directx 11, and shader 2.0 or 3.0, and per my searching, the HD 3000 is supposed to support shader model 4.0 and I definitely have directx 11 installed.... I think I will try and find Stardew's actual support page

the HD 3000 only supports Directx 10.1 shader model 4.1 you should of installed Directx 10 or 10.1 not 11 that you can't use.