The 2012 Mac mini has a HDMI 1.4 port, which tops out at 340 MHz pixel clock. 3440×1440 at 60 Hz requires 319.75 MHz pixel clock using CVT-RB timings, so it should work - technically. The problem is macOS artificially limits the HDMI port to a pixel clock of 165 MHz, so it won't work OOTB and a pixel clock patch of some sort is required.
DisplayPort 1.1a, which the 2012 Mac mini also has, tops out at 360 MHz pixel clock and can thus also do 3440×1440 at 60 Hz. It is, however, possible that macOS artificially limits the DisplayPort output to a pixel clock of 270 MHz (it does on a 2011 Mac mini’s HD 3000) but I've not tested the 2012’s HD 4000 yet), because that is required for 2560×1600 at 60 Hz, i.e. the maximum resolution the Mac mini officially "supports". So, again, a pixel clock patch may be required. FWIW this 270 MHz limit also allows 3840×2160 at 30 Hz, so it's not surprising the 2012 mini can also “do” that.
UPDATE: I have now tested the HD 4000 and it
is subject to the same artificial 270 MHz pixel clock limit as the HD 3000, so 3440×1440 at 60 Hz isn't possible unless the limit is patched.