A proposed high energy Future Circular Hadron-Electron Collider would provide sufficient energy in a clean environment to probe di-Higgs production. Using this channel we show that the azimuthal angle correlation between the missing transverse energy and the forward jet is a very good probe for the non-standard $hhh$ and $hhWW$ couplings. We give the exclusion limits on these couplings as a function of integrated luminosity at a $95$\% C.L. using the fiducial cross sections. With appropriate error fitting methodology we find that the Higgs boson self coupling could be measured to be $g^{(1)}_{hhh} = 1.00^{+0.24(0.14)}_{-0.17(0.12)}$ of its expected Standard Model value at $\sqrt s = 3.5(5.0)$ TeV for an ultimate 10 ab$^{-1}$ of integrated luminosity.