A gene drive method of particular interest for population suppression utilizes homing endonuclease genes (HEGs), wherein a site-specific, nuclease-encoding cassette is copied, in the germline, into a target gene whose loss of function results in loss of viability or fertility in homozygous, but not heterozygous, progeny. Earlier work in Drosophila and mosquitoes utilized HEGs consisting of Cas9 and a single guide RNA (gRNA) that together target a specific gene for cleavage. Homing was observed, but resistant alleles immune to cleavage, while retaining wild-type gene function, were also created through nonhomologous end joining. Such alleles prevent drive and population suppression. Targeting a gene for cleavage at multiple positions has been suggested as a strategy to prevent the appearance of resistant alleles. To test this hypothesis, we generated two suppression HEGs in Drosophila melanogaster targeting genes required for embryonic viability or fertility, using a HEG consisting of CRISPR/Cas9 and gRNAs designed to cleave each gene at four positions. Rates of target locus cleavage were very high, and multiplexing of gRNAs prevented resistant allele formation. However, germline homing rates were modest, and the HEG cassette was unstable during homing events, resulting in frequent partial copying of HEGs that lacked gRNAs, a dominant marker gene, or Cas9. Finally, in drive experiments, the HEGs failed to spread due to the high fitness load induced in offspring as a result of maternal carryover of Cas9/gRNA complex activity. Alternative design principles are proposed that may mitigate these problems in future gene drive engineering.