  We estimate the expected precision at a multi-TeV muon collider for measuring
the Higgs boson couplings with electroweak gauge bosons, $HVV$ and $HHVV\
(V=W^\pm,Z)$, as well as the trilinear Higgs self-coupling $HHH$. At very high
energies both single and double Higgs productions rely on the vector-boson
fusion (VBF) topology. The outgoing remnant particles have a strong tendency to
stay in the very forward region, leading to the configuration of the "inclusive
process" and making it difficult to isolate $ZZ$ fusion events from the $WW$
fusion. In the single Higgs channel, we perform a maximum likelihood analysis
on $HWW$ and $HZZ$ couplings using two categories: the inclusive Higgs
production and the 1-muon exclusive signal. In the double Higgs channel, we
consider the inclusive production and study the interplay of the trilinear
$HHH$ and the quartic $VVHH$ couplings, by utilizing kinematic information in
the invariant mass spectrum. We find that at a centre-of-mass energy of 10 TeV
(30 TeV) with an integrated luminosity of 10 ab$^{-1}$ (90 ab$^{-1}$), one may
reach a 95\% confidence level sensitivity of 0.073\% (0.023\%) for $WWH$
coupling, 0.61\% (0.21\%) for $ZZH$ coupling, 0.62\% (0.20\%) for $WWHH$
coupling, and 5.6\% (2.0\%) for $HHH$ coupling. For dim-6 operators
contributing to the processes, these sensitivities could probe the new physics
scale $\Lambda$ in the order of $1-10$ ($2-20$) TeV at a 10 TeV (30 TeV) muon
collider.
