Apparent channel-dependent violations of the Okubo-Zwieg-Iizuka (OZI) rule in nucleon-antinucleon annihilation reactions in the presence of an intrinsic strangeness component in the nucleon are discussed. Admixture of $s\overline{s}$ quark pairs in the nucleon wave function enables the direct coupling to the $\ensuremath{\phi}$-meson in the annihilation channel without violating the OZI rule. Three forms are considered in this work for the strangeness content of the proton wave function, namely, the $uud$ cluster with a $s\overline{s}$ sea-quark component, kaon-hyperon clusters based on a simple chiral quark model, and the pentaquark picture $uuds\overline{s}$. Nonrelativistic quark model calculations reveal that the strangeness magnetic moment ${\ensuremath{\mu}}_{s}$ and the strangeness contribution to the proton spin ${\ensuremath{\sigma}}_{s}$ from the first two models are consistent with recent experimental data, where ${\ensuremath{\mu}}_{s}$ and ${\ensuremath{\sigma}}_{s}$ are negative. For the third model, the $uuds$ subsystem with the configurations $[31{]}_{FS}[211{]}_{F}[22{]}_{S}$ and $[31{]}_{FS}[31{]}_{F}[22{]}_{S}$ leads to negative values of ${\ensuremath{\mu}}_{s}$ and ${\ensuremath{\sigma}}_{s}$. With effective quark line diagrams incorporating the $^{3}P_{0}$ model, we give estimates for the branching ratios of the annihilation reactions at rest $p\overline{p}\ensuremath{\rightarrow}\ensuremath{\phi}X$ ($X={\ensuremath{\pi}}^{0}$, $\ensuremath{\eta}$, ${\ensuremath{\rho}}^{0}$, $\ensuremath{\omega}$). Results for the branching ratios of $\ensuremath{\phi}X$ production from atomic $p\overline{p}$ $s$-wave states are for the first and third model found to be strongly channel dependent, in good agreement with measured rates.