A search for squarks of $R$-parity violating supersymmetry is performed in $ep$ collisions at HERA using H1 1994$
ightarrow$1997 data corresponding to an integrated luminosity of $37 \picob^{-1}$. Direct single production of squarks of each generation by $e^+$-quark fusion via a Yukawa coupling $\lambda'$ is considered and all possible $R$-parity violating decays and gauge decays of the squarks are taken into account. The dependence of the sensitivity to $\lambda'$ on the free parameters of the Minimal Supersymmetric Standard Model (MSSM) is studied in detail. The reach in the mass-coupling plane extends to domains unexplored in other direct or indirect searches. For a Yukawa coupling of the electromagnetic strength squark masses below $258 \GeV$ can be excluded at the $95 \%$ confidence level independently of the values of the MSSM parameters. This sensitivity still reaches $182 \GeV$ for a 100 times smaller coupling strength. In addition, for a fixed value of the Yukawa coupling $\lambda'$, HERA results are interpreted for the first time in terms of constraints on the parameters of the Minimal Supergravity model.