We analyse the sensitivity of the proposed Compact Linear Collider (CLIC) to the existence of beyond the Standard Model (SM) Higgs bosons through their decays into pairs of massive gauge bosons $H to VV$ and SM-like Higgses $H to hh$, considering CLIC centre of mass energies $sqrt{s} = 1.4$ TeV and $3$ TeV. We find that resonant di-Higgs searches at CLIC would allow for up to two orders of magnitude improvement w.r.t. the sensitivity achievable by HL-LHC in the mass range $m_H in [250,mathrm{GeV},, 1 ,mathrm{TeV}]$. Focusing then on a real singlet extension of the SM, we explore the prospects of heavy Higgs searches at CLIC for probing the regions of parameter space yielding a strongly first order electroweak phase transition that could generate the observed matter-antimatter asymmetry of the Universe. Our study illustrates the complementarity between CLIC and other possible future colliders like FCC-ee in probing singlet extensions of the SM, and shows that high-energy $e^+ e^-$ colliders provide a powerful means to unravel the nature of electroweak symmetry breaking in the early Universe.
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