In this paper, we first synthesized an amphiphilic block copolymer poly(ethylene glycol)-poly(2,4,6-trimethoxybenzylidene-pentaerythritol carbonate-co-5-methyl-5-propargyl-1,3-dioxan-2-one), i.e., PEG-P(TMBPEC-co-MPMC), with pendant reactive alkynyl groups as well as pH-sensitive acetal groups. Next, core crosslinked (CCL) micelles were prepared by the introduction of 1,6-diazidohexane and bis(azidoethyl)disulfide into micelles via azide–alkyne click chemistry, which were denoted as CCL/CC and CCL/SS, respectively. The CCL micelles had superior stability and drug loading efficiency to the uncrosslinked (UCL) micelles. In comparison with free DOX, drug-loaded CCL micelles exhibited lower cell viability in MCF-7/ADR cells due to their “stealth” endocytosis effect that might be beneficial for overcoming delivery barriers of drug resistance. More interestingly, as compared with CCL/CC micelles, CCL/SS micelles were found to further enhance cytotoxicity in MCF-7/ADR cells because of their better on-demand drug release capability of pH and redox dual-sensitive CCL/SS micelles. These results suggest that the self-assembled pH and redox dual-sensitive CCL/SS micelles have promising applications to overcome multi-drug resistance in tumor treatments.